mil 


LltfKAKY 


HANDBOOK 


OF   THE 


STEAM-ENGINE 


CONTAINING     ALL     THE     RCLES     REQUIRED     FOR     THE     RIGHT 

CONSTRUCTION    AND    MANAGEMENT   OF  ENGINES    OF  EVERY   CLASS,  WITH 

THE  EASY  ARITHMETICAL  SOLUTION  OF  THOSE  RULES. 


CONSTIirXIXG 


A    KEY 


CATECHISM  OF  THE  STEAM-ENGINE.' 


ILLUSTRATED   BY 

SIXTY-SEVEN    WOOD-CUTS,   AND   NUMEROUS   TABLES   AND    EXAMPLES. 


BY 

JOHN  BOURNE,  C.E., 

AtmiOK    OF    'A    TREATISE    ON    THE    STEAM-ENGINE,'    'A     TKEATISB    ON    THl 
8CMW-PKOPELLEB,1   'A  CATECHISM   OF  THE   STKAM-K.NGINK,1   ETC. 


NEW  YORK : 
D.    APPLETON    AND     COMPANY, 

448    &    445    BROADWAY. 
1865. 


TO 


GEOEGE  TUMBULL,  ESQ.,  'C.E.,  F.R.A.S.,  ETC., 


LATE   ENGINEER-IN-CHIEF  OF  THE  EAST  INDIAN  RAILWAY. 


MY   DEAE  MB.  TUBNBULL, 

In  dedicating  the  present  Work  to  you  I  am  moved  by 
two  main  considerations : — First,  to  testify  in  the  best  manner 
I  can  my  regard  and  esteem  for  you  personally ;  and  Second^  to 
mark  my  sense  of  the  skill,  tact,  and  abiding  integrity  which 
you  brought  to  the  onerous  duty  of  constructing  the  first  and 
greatest  of  "the  Indian  railways,  and  of  which,  while  in  India,  I 
had  opportunities  of  forming  a  just  appreciation. 

The  public  in  this  country — traditionally  so  ignorant  of 
India — has  yet  to  learn  the  important  fact,  that  the  works 
carried  out  under  your  direction  in  that  country,  are  greater 
and  more  difficult  than  most  of  those  which  are  to  be  found  at 
home ;  and  that  among  other  achievements,  you  constructed 
the  largest  bridge  in  the  world — the  great  bridge  over  the  St. 
Lawrence  alone  excepted.  But  these  technical  successes,  im- 
portant as  they  are,  were  not  more  eminent  than  those  which 
you  won  over  the  discouragements  and  difficulties  of  the  Indian 
official  system — ending,  too,  in  gaining  the  esteem  and  appro- 


IV  DEDICATION. 

bation  of  the  Indian  Government,  as  well  as  of  those  for  whom 
you  zealously  labored  for  so  many  years  in  India. 

Whatever  the  benefits  may  be  of  the  Indian  railways,  their 
greatest  benefit  is  that  they  have  taken  to  that  country  men  who 
have  impressed  the  people  with  their  skill,  and  who  have  ac- 
quired an  accurate  perception  of  the  physical  wants  of  the 
country,  together  with  all  that  practical  knowledge  of  localities 
which  will  enable  them  to  carry  out  with  confidence,  economy, 
and  success,  the  numerous  improvements  still  required  by  that 
great  dependency,  and  upon  which  only  a  comparatively  small 
beginning  has  yet  been  made. 

I  remain,  my  dear  Mr.  Turnbull, 

Truly  yours, 

J.  BOUENE. 


PREFAC  E. 


THE  present  work,  designed  mainly  as  a  Key  to  my 
'  Catechism  of  the  Steam-Engine,'  has,  during  its  compo- 
sition, been  somewhat  extended  in  its  scope  and  objects,  so 
as  also  to  supply  any  points  of  information  in  which  it 
appeared  to  me  the  Catechism  was  deficient,  or  whereby 
the  utility  of  this  Handbook  as  a  companion  volume  would 
be  increased. 

The  purpose  of  the  Catechism  being  rather  to  enun- 
ciate sound  principles  than  to  exemplify  the  application  of 
those  principles  to  practice,  it  was  always  obvious  to  me 
that  another  work  which  would  point  out  in  the  plainest 
possible  manner  the  methods  of  procedure  by  which  all 
computations  connected  with  the  steam-engine  were  to  be 
performed — illustrated  by  practical  examples  of  the  appli- 
cation of  the  several  rules — was  indispensable  to  satisfy 


VI  PREFACE. 

the  wants  of  the  practical  engineer  in  this  department 
of  enquiry.  The  present  work  was  consequently  begun, 
and  part  of  it  was  printed,  several  years  ago,  but  the 
pressure  of  other  pursuits  has  heretofore  hindered  its 
completion ;  and  in  now  sending  it  forth  I  do  so  with  the 
conviction  that  I  have  spared  no  pains  to  render  it  as 
useful  as  possible  to  the  large  class  of  imperfectly  educated 
engineers  to  whom  it  is  chiefly  addressed.  It  is  with  the 
view  of  enabling  its  expositions  to  be  followed  by  those 
even  of  the  most  slender  scientific  attainments  that  I  have 
introduced  the  first  chapter,  explaining  those  several  pro- 
cesses of  arithmetic  by  which  engineering  computations 
are  worked  out.  For  although  there  is  no  want  of  man- 
uals imparting  this  information,  there  are  none  of  them, 
that  I  know  of,  which  have  special  reference  to  the  wants 
of  the  engineer ;  and  none  of  them  deal  with  those  asso- 
ciations, by  way  of  illustration,  with  which  the  engineer 
is  most  familiar.  Indeed,  engineers,  like  sailors  and  other 
large  classes  of  men,  have  an  order  of  ideas,  and,  to  some 
extent,  even  a  species  of  phraseology  of  their  own ;  and 
the  avenues  to  their  apprehension  are  most  readily  opened 
by  illustrations  based  upon  their  existing  knowledge  and 
experience,  such  as  an  engineer  can  best  supply.  By  this 
familiar  method  of  exposition  the  idea  of  difficulty  is  dis- 


PREFACE.  Vll 

pelled  ;  and  science  loses  half  its  terrors  by  losing  all  its 
mystery. 

If  I  might  infer  the  probable  reception  of  the  present 
work  from  the  numerous  anxious  enquiries  addressed  to 
me  from  all  quarters  of  the  world  during  the  last  ten 
years,  touching  the  prospects  of  its  speedy  appearance,  I 
should  augur  for  it  a  wider  popularity  than  any  work  I 
have  yet  written.  The  questions  propounded  to  me  by 
engineers  and  others,  in  consequence  of  the  offer  I  made 
in  the  preface  to  my  '  Catechism  of  the  Steam-Engine,'  in 
1856,  to  endeavour  by  my  explanations  to  remove  such 
difficulties  as  impeded  their  progress,  have  had  the  effect 
of  showing  more  clearly  than  I  could  otherwise  have  per- 
ceived what  the  prevalent  difficulties  of  learners  have 
been  ;  and  I  have  consequently  been  enabled  to  give  such 
explanations  in  the  present  work  as  appeared  best  calcu- 
lated to  meet  those  difficulties  for  the  future. 

To  several  of  my  correspondents  I  have  to  acknowledge 
myself  indebted  for  the  correction  of  typographical  errors 
in  my  several  works,  and  also  for  valuable  suggestions  of 
various  kinds,  which  I  have  made  use  of  in  every  case  in 
which  they  were  available. 

I  may  here  take  occasion  to  notify  that  I  have  lately 
prepared  an  Introduction  to  my  '  Catechism  of  the  Steam- 


Vlll  PREFACE. 

Engine,'  which  reviews  the  most  important  improvements 
of  the  last  ten  years ;  and  which,  for  the  convenience  of 
persons  already  possessing  the  Catechism,  may  be  had 
separately.  These  three  works  taken  together  form  a  body 
of  engineering  information  so  elementary  as  to  be  intelli- 
gible by  anybody,  and  yet  so  full  that  the  attentive  student 
of  them  will,  I  trust,  be  found  not  to  fall  far  short  of  the 
most  proficient  engineers  in  all  that  relates  to  a  knowledge 
of  the  steam-engine  in  its  most  important  applications. 

J.  BOUENE. 

BERKELEY  VILLA,  REGENT'S  PAKK  ROAD, 
LONDON:  1865. 


CONTENTS. 


CHAPTER  I. 

ARITHMETIC    OP   THE    STEAM-ENGINE. 

PAGE 

Principles  of  Numeration          ......        1 

Addition        ........  10 

Subtraction         ........      13 

Multiplication  .......  16 

Division  .........       24 

Nature  and  Properties  of  Fractions  ....  30 

Addition  and  Subtraction  of  Fractions  .  .  .  .34 

To  Reduce  Fractions  to  a  Common  Denominator  .  .  35 

Multiplication  and  Division  of  Fractions         .  .  .  .38 

Proportion,  or  Rule  of  Three          .....  42 

Squares  and  Square  Roots  of  Numbers         .  •         «*  .44 

Cubes  and  Cube  Roots  of  Numbers  .  .  .  .  48 

On  Powers  and  Roots  in  General         .  .  .  .  .49 

Roots  as  represented  by  Fractional  Exponents    ...  51 

Logarithms         ........      52 

Compound  Quantities          ......  57 

Resolution  of  Fractions  into  Infinite  Series  .  .  .66 

Equations      .........  74 

CHAPTER  II. 

MECHANICAL   PRINCIPLES    OF   THE  STEAM-ENQINE. 

Law  of  the  Conservation  of  Force       .  .  .  .  .78 

Law  of  Virtual  Velocities    .  .  .  .  .  79 

Nature  of  Mechanical  Power    .  90 


X  CONTENTS. 

PAGE 

Mechanical  Equivalent  of  Heat.      .....  91 

Laws  of  Falling  Bodies  .  .  .  .  .  .93 

Motion  of  Fluids       .......          100 

Inertia  and  Momentum  ......     105 

Centrifugal  Force     .......          107 

Bodies  Revolving  in  a  Circle    ......     107 

Centres  of  Gyration  and  Percussion          ....          112 

The  Pendulum    ........     114 

The  Governor  .......          116 

Friction   .........     118 

Strength  of  Materials          ......          124 

Strength  of  Pillars,  Beams,  and  Shafts          .  .  .  .128 

CHAPTER  III. 

THEORY   OF  THE   STEAM-ENGINE. 

Nature  and  Effects  of  Heat      .  .  .  .  .  .134 

Difference  between  Temperature  and  Quantity  of  Heat  .         136 

Absolute  Zero     ........    136 

Fixed  Temperatures  ......          137 

Thermometers    ........    137 

Dilatation      ........          140 

Liquefaction       ........    150 

Vaporisation  .......          152 

Pressure  of  Steam  at  Different  Temperatures  .  .  .    157 

Specific  Heat .162 

Phenomena  of  Ebullition          .  .  .  .  .  .168 

Communication  of  Heat      ......          171 

Combustion      ' .  .  .  .  .  .  .     174 

Thermodynamics      .......          180 

Expansion  of  Steam     .  .         •>»f~'        ....     182 

Velocity  and  Friction  of  Running  Water  .  .  ...         199 

CHAPTER  IV. 

PROPORTIONS   OP  STEAM-ENGINES. 

Nominal  Power              .           .           .           .           .  .           .    208 

General  Proportions            .         .k  A;      «           .  •           .212 

Steam  Ports        .        >-^           .           .           ,  _  .  |      •    216 

Steam  Pipe    .            .            .                   '  ~;  !    ''"  ^ '  -"  .            .          218 

Safety  Valves     ....      >fu»uiC      .  .           .219 

Feed  Pipe      ....           ».-M      .  .                     221 


HANDBOOK 


THE    STEAM-ENGINE. 

CHAPTER  I. 

ARITHMETIC  OF  THE  STEAM-ENGINE. 

IN  this  chapter  I  propose  to  explain  as  plainly  and  simply  as  I 
can  those  principles  of  arithmetic  which  it  is  necessary  to  know, 
that  we  may  he  ahle  to  perform  all  ordinary  engineering  calcula- 
tions. In  order  that  my  remarks  may  he  generally  useful  to  work- 
ing mechanics  of  little  education,  I  shall  proceed  upon  the  suppo- 
sition that  the  reader  is  not  merely  destitute  of  all  arithmetical 
knowledge,  hut  that  he  has  no  ideas  of  number  or  quantity  that 
are  not  of  the  most  vague  and  indefinite  description.  I  have  known 
many  engineers — who  were  otherwise  men  of  ahility — to  be  in 
this  condition  ;  and  the  design  of  these  observations  is  to  enable 
such,  with  the  aid  of  their  own  common  sense  and  their  familiar 
associations,  to  arrive  at  tangible  ideas  respecting  the  properties  of 
numbers,  and  to  perform  with  facility  all  the  ordinary  engineering 
calculations  which  occur  in  the  requirements  of  engineering  prac- 
tice. These  various  topics  are  not  beset  with  any  serious  diffi- 
culty. The  processes  of  arithmetic  are  merely  expedients  for  faci- 
litating the  discovery  of  results  which  every  mechanic  of  ordinary 
ingenuity  would  find  a  means  of  discovering  for  himself,  if 
really  called  upon  to  set  about  the  task ;  and  it  is  mainly  b£- 
1 


2  ARITHMETIC    OF   THE    STEAM-ENGINE. 

cause  the  rationale  of  these  processes  has  not  been  much  ex- 
plained in  school  treatises,  hut  the  results  presented  as  feats  of 
legerdemain  performed  by  the  application  of  a  certain  rule — the 
reason  of  which  is  not  made  apparent — that  the  idea  of  difficulty 
has  arisen  in  connection  with  such  enquiries.  The  rudest  and 
most  savage  nations  have  all  some  species  or  other  of  arithmetic 
suited  to  their  requirements.  The  natives  of  Madagascar,  when 
they  wish  to  count  the  nuniber  of  men  in  their  army,  cause  the 
men  to  proceed  through  a  narrow  pass,  where  they  deposit  a 
stone  for  each  man  that  goes  through ;  and  hy  subsequently  ar- 
ranging these  stones  in  groups  of  ten  each,  and  these  again  in 
groups  of  a  hundred,  and  so  on,  they  are  enabled  to  arrive  at  a 
precise  idea  of  the  number  of  men  the  army  contains.  A  la- 
bourer in  counting  bricks  out  of  a  cart  or  barge  makes  a  chalk- 
mark  on  a  board  for  every  ten  bricks  he  hands  out ;  and  these 
chalk-marks  he  arranges  in  groups  of  five  or  ten  each,  so  that 
he  may  easily  reckon  up  the  total  number  of  groups  the  board 
contains.  These  are  expedients  of  numeration  which  the  most 
moderate  intelligence  will  suggest  as  conducive  to  the  acquisi- 
tion of  the  idea  of  quantity ;  and  the  rules  of  arithmetic  are 
merely  an  extension  and  combination  of  such  methods  as  expe- 
rience has  shown  to  be  the  most  convenient  in  practice  to  ac- 
complish the  ends  sought. 

It  will  be  obvious  that  the  number  of  stones  or  chalk-marks 
collected  into  groups  in  the  preceding  examples  may  either  be 
five,  ten,  twelve,  or  any  other  number ;  the  only  necessary  con- 
dition being  that  the  number  in  each  group  shall  be  the  same. 
The  concurrent  practice  of  most  nations,  however,  is  to  employ 
groups  consisting  of  ten  objects  in  each  group ;  no  doubt  from 
the  circumstances  that  mankind  are  furnished  with  ten  fingers, 
and  because  the  fingers  are  much  used  in  most  primitive  systems 
of  numeration.  In  some  cases,  however,  objects  are  reckoned 
by  the  dozen,  or  score,  or  gross  ;  or,  in  other  words,  a  dozen,  a 
score,  or  a  gross  of  objects  are  collected  in  each  group.  But  in 
the  ordinary  or  decimal  system  of  numeration,  ten  objects  or 
units  are  supposed  to  be  collected  in  each  group,  and  ten  of  these 
primary  groups  are  supposed  to  be  collected  in  each  higher  or 


DIFFERENT   EXPEDIENTS    OF   NTJMERATION.  3 

larger  group  of  the  class  immediately  above,  and  so  on  indefi- 
nitely. The  decimal  system  is  so  called  from  the  Latin  word 
dccem,  signifying  ten,  and  the  word  unit  is  derived  from  the 
Latin  word  unus,  signifying  one.  Ten  units  form  a  group  of  ten, 
and  ten  of  these  groups  form  a  group  of  a  hundred,  and  ten  groups 
of  a  hundred  form  a  group  of  a  thousand,  and  so  on  for  ever. 

The  Romans,  whose  numbers  are  still  commonly  used  on  clock 
faces,  employed  a  mark  or  i  to  signify  one ;  two  marks  or  n  to 
signify  two ;  three  marks  or  in  to  signify  three ;  and  four  marks 
or  im  to  signify  four.  But  as  it  would  have  been  difficult  to 
count  these  marks  if  they  became  very  numerous,  they  employed 
the  letter  v  to  signify  five  and  the  letter  x  or  a  cross  to  signify 
ten,  and  v  is  the  same  mark  as  one-half  of  x,  which  was  no  doubt 
the  primary  of  the  two  characters.  An  i  appended  to  the  left- 
hand  side  of  the  v  or  x  signified  v  or  x  diminished  by  one, 
whereas  each  additional  i  added  to  the  right-hand  side  of  the  v 
or  x,  signified  one  added  to  v  or  x.  Thus  according  to  the 
Eoman  numeration  rv  signifies  four ;  vi  signifies  six ;  ix  signifies 
nine ;  xi  signifies  eleven ;  xn  signifies  twelve ;  and  so  on.  A 
hundred  is  signified  by  the  letter  o,  the  initial  letter  of  the  Latin 
word  centum,  signifying  a  hundred;  and  a  thousand  is  repre- 
sented by  the  letter  M,  the  initial  letter  of  the  Latin  word  mille, 
signifying  a  thousand. 

It  is  clear  that  the  Eoman  numeration,  though  adequate  to  the 
wants  of  a  primitive  people,  was  a  very  crude  and  imperfect  sys- 
tem. It  has  therefore  been  long  superseded  for  all  arithmetical 
purposes  by  the  system  of  notation  at  present  in  common  use, 
and  which  has  a  distinct  sign  or  figure  for  each  number  up  to  9, 
and  a  cipher  or  0,  which  has  no  individual  value,  but  which  af- 
fects the  value  of  other  figures.  This  system,  which  came  origi- 
nally from  India,  was  brought  into  Europe  by  the  Moors ;  and  in 
common  with  most  of  the  oriental  languages,  it  is  written  from 
right  to  left  instead  of  from  left  to  right,  like  the  languages  of 
Europe,  so  that  in  performing  a  sum  in  arithmetic — as  in  writing 
a  word  in  Sanscrit  or  Arabic — we  have  to  begin  at  the  right- 
hand  side  of  the  page.  In  this  system  the  classes  or  orders  of 
the  objects  or  groups  of  objects  is  indicated  by  the  place  occu- 


4  ARITHMETIC    OF   THE   STEAM-ENGINE. 

pied  by  the  figures  which  express  their  value.  Thus  in  the  case 
of  the  groups  of  stones  employed  in  Madagascar,  the  figure  3 
may  he  employed  to  designate  either  three  individual  stones,  or 
three  groups  of  ten  each,  or  three  groups  of  a  hundred  each ;  hut 
in  using  the  figure  it  is  quite  indispensable  that  it  should  appear, 
by  some  distinctive  mark,  which  order  or  class  is  intended  to  be 
designated.  "We  might  use  the  figure  3  to  designate  three  single 
stones,  and  we  might  use  the  figure  with  a  circle  round  it  to  de- 
note groups  of  ten  each,  and  with  a  square  round  it  to  denote 
groups  of  a  hundred  each.  But  on  trial  of  such  a  system  we 
should  find  it  to  be  very  cumbrous  and  perplexing,  and  the  method 
found  to  be  most  convenient  is  to  add  a  cipher  after  the  three  to 
show  that  groups  of  tens  are  intended  to  be  signified,  and  two 
ciphers  to  show  that  groups  of  hundreds  are  intended  to  be  sig- 
nified. Three  groups  of  tens,  or  thirty,  are  therefore  expressed 
by  30,  and  three  groups  of  hundreds  are  expressed  by  300. 
Here  the  ciphers  operate  wholly  in  advancing  the  3  into  a  higher 
and  higher  position,  which,  however,  other  figures  will  equally 
suffice  to  do  if  there  are  any  such  to  be  expressed.  Three  groups 
of  one  hundred  stones  in  each,  three  groups  of  ten  stones  in  each, 
and  three  individual  stones,  will  therefore  be  represented  by  the 
number  333,  in  which  the  same  figure  recurs  three  times,  but 
which  is  counted  ten  times  greater  at  each  successive  place  to 
which  it  is  advanced,  reckoning  from  the  right  to  the  left.  Of 
course,  the  number  three  hundred  and  thirty-three  might  be 
represented  in  an  infinite  number  of  other  ways,  differing  more 
or  less  from  the  one  here  indicated ;  and  any  of  the  properties 
belonging  to  the  number  would  equally  hold  by  whatever 
expedient  of  notation  it  was  expressed.  But  the  manner  here 
described  is  that  which  the  accumulated  experience  of  mankind 
has  shown  to  be  the  most  convenient;  and  it  is  therefore  gen- 
erally adopted,  though  it  is  proper  to  understand  that  there  is  no 
more  necessary  relation  between  the  number  itself  and  the  com- 
mon mode  of  expressing  it,  than  there  is  between  the  Latin  word 
eqmis,  a  horse,  and  that  most  useful  of  quadrupeds.  In  each 
case  the  relations  are  wholly  conventional,  and  might  be  altered 
without  in  any  way  affecting  the  object. 


NATURE   OF   ARITHMETIC.  5 

Arithmetic  is  the  science  of  numbers.  Numbers  treat  of 
magnitude  or  quantity ;  and  whatever  is  capable  of  increase  or 
diminution  is  a  magnitude  or  quantity.  A  sum  of  money,  a 
weight,  or  a  surface,  is  a  quantity,  being  capable  of  increase  or 
diminution.  But  as  we  cannot  measure  or  determine  any  quan- 
tity except  by  considering  some  other  quantity  of  the  same  kind 
as  known,  and  pointing  out  their  mutual  relation,  the  measure- 
ment of  quantity  or  magnitude  of  ah1  kinds  is  reduced  to  this :  fix  at 
pleasure  upon  any  one  known  kind  of  magnitude  of  the  same  spe- 
cies as  that  which  has  to  be  determined,  and  consider  it  as  the 
measure  or  unit,  and  determine  the  proportion  of  the  proposed 
magnitude  to  this  known  measure.  This  proportion  is  always 
expressed  by  numbers ;  so  that  number  is  nothing  more  than  the 
proportion,  of  one  magnitude  to  that  of  some  other  magnitude 
arbitrarily  assumed  as  the  unit.  If,  for  example,  we  want  to 
determine  the  magnitude  of  a  sum  of  money,  we  must  take  some 
piece  of  known  value — such  as  the  pound  or  shilling — and  show 
how  many  such  pieces  are  contained  in  the  given  sum.  If  we 
wish  to  express  the  distance  between  two  cities,  we  must  use 
some  such  recognized  measure  of  length  as  the  foot  or  mile ;  and 
if  we  wish  to  ascertain  the  magnitude  of  an  estate,  we  must  em- 
ploy some  such  measure  of  surface  as  the  square  mile  or  acre. 
The  foot-rule  is  the  measure  of  length  most  used  for  engineering 
purposes.  The  foot  is  divided  into  twelve  inches,  and  the  inch 
is  subdivided  into  half  inches,  quarter  inches,  eighths,  and  six- 
teenths. It  is  clear  that  two  half  inches  or  four  quarter  inches 
make  an  inch,  as  also  do  eight  eighths  and  sixteen  sixteenths ; 
and  indeed  it  is  obvious  that  into  whatever  number  of  parts  the 
inch  is  divided,  we  shall  equally  have  the  whole  inch  if  we  take 
the  whole  of  the  parts  of  it.  If  the  inch  were  to  be  divided  into 
ten  equal  parts,  then  ten  of  these  parts  would  make  an  inch. 
Fractional  parts  of  an  inch,  or  of  any  other  quantity,  are  ex- 
pressed as  follows :  a  half,  -J-;  a  quarter,  £;  an  eighth,  -J- ;  a  six- 
teenth, 7V ;  and  a  tenth,  TV  The  figure  above  the  line  is  called 
the  numerator,  because  it  fixes  the  number  of  halves,  quarters, 
or  eighths,  which  is  intended  to  be  expressed ;  and  the  figure 
below  the  line  is  called  the  denominator,  because  it  fixes  the 


6  AKITHMETIC    OF   THE    STEAM-ENGINE. 

order  or  denomination  of  the  fraction,  whether  halves,  quarters, 
eighths,  or  otherwise.     Thus  in  the  fractions  f  ths  and  Iths,  the 
figures  3  and  7  are  the  numerators,  and  the  figures  4  and  8  the 
denominators;  and  fths,  |ths,  or  }jjths,  are  clearly  equal  to  1. 
So  also  |ths,  ^ths,  and  jgths  are  clearly  greater  than  1,  the  first 
being  equal  to  1-Jth,  the  second  to  Hth,  and  the  third  to  lTlffth. 
The  species  of  fractions  here  referred  to  is  that  which  is 
called  vulgar  fractions,  as  being  the  kind  of  fractions  in  common 
use ;  and  every  engineer  who  speaks  of  f  ths  or  fths  of  an  inch, 
and  every  housewife  who  speaks  of  f  of  a  pound  of  sugar,  or  ^  a 
pound  of  tea,  refers,  perhaps  unconsciously,  to  this  species  of 
numeration.     There  is  another  species  of  fractions,  however, 
called  decimal  fractions,  not  usually  employed  for  domestic  pur- 
poses, but  which  is  specially  useful  in  arithmetical  computations, 
and  these  fractions  being  dealt  with  in  precisely  the  same  man- 
ner as  ordinary  figures,  are  very  easy  in  their  application.     In 
ordinary  figures,  the  value  of  each  succeeding  figure,  counting 
from  the  right  to  the  left,  is  ten  times  greater  than  the  preceding 
one,  in  consequence  of  its  position ;  and  in  decimal  fractions  the 
value  of  each  succeeding  figure,  counting  from  left  to  right,  is 
ten  times  less.    Thus  the  figures  1111  signify  one  thousand  one 
hundred  and  eleven ;  and  if  after  the  last  unit  we  place  a  period 
or  full  stop,  and  write  a  one  after  it  thus,  llll'l,  we  have  one 
thousand  one  hundred  and  eleven  and  one-tenth.    The  period, 
or  decimal  point,  as  it  is  termed,  prefixed  to  any  number,  im- 
plies that  it  is — not  a  whole  number — but  a  decimal  fraction. 
Thus  '1  means  one-tenth,  •£  two-tenths,  *3  three-tenths,  -4  four- 
tenths,  and  so  on.    So  in  like  manner  '11  means  one-tenth  and 
one  hundredth,  or  eleven  hundredths ;  '22  means  two-tenths  and 
two  hundredths,  or  twenty-two  hundredths;  "33,  three-tenths 
and  three  hundredths,  or  thirty-three  hundredths ;  and  so  on — 
each  successive  figure  of  the  fraction  counting  from  the  left  to 
the  right,  being  from  its  position  ten  times  less  than  that  which 
went  before  it.    The  number  '1111  signifies  one  thousand  one 
hundred  and  eleven  ten  thousandths,  the  first  decimal  place 
being  tenths,  the  next  hundredths,  the  next  thousandths,  the 
next  ten  thousandths,  and  so  on.    If  we  wish  to  express  a  hun- 


NATURE    OF   DECIMAL   FRACTIONS.  7 

dredth  by  this  notation,  we  place  a  cipher  before  the  unit  thus, 
•01 ;  if  a  thousandth  two  ciphers,  -001 ;  and  so  of  all  other  quan- 
tities. The  multiplication,  division,  and  all  the  other  arithmeti- 
cal operations  required  to  be  performed  with  decimal  fractions, 
are  conducted  in  precisely  the  same  manner  as  if  they  were 
ordinary  numbers — the  decimal  progression  being  carried  down- 
wards in  the  one  case  precisely  in  the  same  manner  as  it  is  car- 
ried upwards  in  the  other  case ;  and  it  is  easy  to  suppose  that 
the  stones  used  by  the  natives  of  Madagascar  may  not  only  be 
collected  into  groups  of  tens  and  hundreds,  but  that  each  stone 
may  also  be  subdivided  into  tenths,  hundredths,  or  thousandths, 
so  that  parts  of  a  stone  may  be  reckoned.  Instead  of  dividing 
the  stone  into  halves,  and  quarters,  and  eighths,  and  sixteenths, 
as  would  be  done  by  the  method  of  vulgar  fractions,  it  is  sup- 
posed by  the  decimal  system  of  fractions  to  be  at  once  divided 
into  tenths,  whereby  the  same  system  of  grouping  by  tens,  which 
is  used  above  unity,  is  also  rendered  applicable  to  the  fractional 
parts  below  unity — to  the  great  simplification  of  arithmetical 
processes.  In  all  cases  a  decimal  fraction  may  be  transformed 
into  a  vulgar  fraction  of  equal  value  by  retaining  the  significant 
figures  as  the  numerator,  and  by  using  as  the  denominator  1, 
with  as  many  ciphers  as  there  are  figures  after  the  decimal  point. 
Thus  •!  is  equal  to  y1^ ;  '11  is  equal  to  —^ ;  '01  is  equal  to  T£ff ; 
•001  is  equal  to  y^J  3-1459  is  equal  to  3  TWffV;  and  '^854  is 
equal  to  TV&V 

In  all  countries  there  are  certain  recognised  standards  of 
magnitude  for  measuring  other  magnitudes  by ;  such  as  the  inch, 
foot,  yard,  or  mile  for  measuring  lengths;  the  square  inch, 
square  yard,  or  square  mile,  or  square  pole,  rood,  or  acre,  for 
measuring  surfaces ;  the  grain,  ounce,  pound,  or  ton  for  measur- 
ing weights ;  and  the  penny,  shilling,  and  sovereign  for  measur- 
ing money.  It  is,  of  course,  quite  inadmissible  in  conducting 
any  of  the  operations  of  arithmetic  to  confound  these  different 
kinds  of  magnitudes  together,  and  there  is  as  much  difference 
between  a  linear  foot  and  a  square  foot  as  there  is  between  a 
ton  weight  and  a  pound  sterling.  A  square  surface  measuring 
an  inch  long  and  an  inch  broad  is  a  square  inch.  A  strip  of  sur- 


8  ARITHMETIC    OF   THE    STEAM-ENGINE. 

face  1  inch  broad  and  12  inches  or  1  foot  long  will  be  equal  to 
12  square  inches ;  and  12  such  strips  laid  side  by  side,  and  there- 
fore a  foot  long  and  a  foot  broad,  will  make  12  times  12  square 
inches,  or  144  square  inches.  In  each  square  foot,  therefore, 
there  are  144  square  inches ;  and  as  there  are  3  linear  feet  in  a 
linear  yard,  there  will  be  in  a  square  yard  9  square  feet,  as  we 
may  suppose  the  square  yard  to  be  composed  of  three  strips  of 
surface,  each  3  feet  long  and  1  foot  wide,  and  therefore  contain- 
ing 3  square  feet  in  each. 

A  cubic  inch  is  a  cube  or  dice  measuring  1  inch  long,  1  inch 
broad,  and  1  inch  deep.  A  square  foot  of  board  1  inch  thick  will 
consequently  make  144  cubic  inches  or  dice  if  cut  up.  But  as  it 
will  take  twelve  such  boards  placed  upon  one  another  to  make 
a  foot  in  depth,  or,  in  other  words,  to  make  a  cubic  foot,  it 
follows  that  there  will  be  12  times  144,  or,  in  all,  1,^18  cubic 
inches  in  the  cubic  foot.  So,  in  like  manner,  as  there  are  3  lin- 
ear feet  in  the  linear  yard,  and  9  square  feet  in  the  square  yard, 
there  will  be  3  times  9  or  27  cubic  feet  in  the  cubic  yard — the 
cubic  yard  being  composed  of  three  strata  1  foot  thick,  contain- 
ing 9  cubic  feet  in  each. 

Besides  the  square  inch  there  is  the  circular  inch  by  which 
surfaces  are  sometimes  measured.  The  circular  inch  is  a  circle 
1  inch  in  diameter,  and  as  it  is  a  fundamental  rule  in  geometry 
that  the  area  of  different  circles  is  proportional  to  the  squares 
of  their  respective  diameters,  the  area  of  any  piston  or  safety- 
valve  or  other  circular  orifice  will  be  at  once  found  in  circular 
inches  by  squaring  its  diameter,  as  it  is  called;  or,  in  other 
words,  by  multiplying  the  diameter  of  such  piston  or  orifice  ex- 
pressed in  inches  by  itself.  Thus  as  a  square  foot,  or  a  square 
of  12  inches  each  way,  contains  144  square  inches,  so  a  circular 
foot,  or  a  circle  of  12  inches  diameter,  contains  144  circular 
inches.  There  is  a  constant  ratio  subsisting  between  a  circular 
inch  or  foot  and  the  square  circumscribed  around  it.  The  cir- 
cular inch  or  foot  is  less  than  the  square  inch  or  foot  by  a  cer- 
tain uniform  quantity ;  and  this  relation  being  invariable,  it  be- 
comes easy  when  we  know  the  area  of  any  circle  in  circular 
inches  to  tell  what  the  equivalent  area  will  be  in  square  inches, 


SQUAEE,  CIRCULAK,  CUBIC,  AND  OTHER  INCHES. 

as  we  have  only  to  multiply  by  a  certain  number — which  will 
be  less  than  unity — in  order  to  give  the  equivalent  area.  This 
number  will  be  a  little  more  than  f ,  or  it  will  be  the  decimal 
•7854 ;  and  if  circular  inches  be  multiplied  by  this  number,  we 
shall'  have  the  same  area  expressed  in  square  inches.  Multiplying 
any  quantity  by  a  number  less  than  unity,  it  may  be  here  re- 
marked, diminishes  the  quantity,  just  as  multiplying  by  a  num- 
ber greater  than  unity  increases  it.  To  multiply  by  •§•  gives  the 
same  result  as  to  divide  by  2  ;  and  to  multiply  by  the  decimal 
•V854:  will  have  the  effect  of  reducing  the  number  by  nearly  a 
fourth,  as  it  is  necessary  should  be  done  in  order  to  convert  cir- 
cular into  square  inches ;  for,  seeing  that  the  square  inches  are 
the  larger  of  the  two,  there  must  be  fewer  of  them  in  any  given 
area. 

Besides  the  cubic  inch  there  are  the  spherical,  the  cylindri- 
cal, and  the  conical  inch,  all  having  definite  relations  to  one 
another.  The  spherical  inch  is  a  ball  an  inch  in  diameter ;  the 
cylindrical  inch  is  a  cylinder  an  inch  in  diameter  and  an  inch 
high ;  and  the  conical  inch  is  a  cone  whose  base  is  an  inch  in 
diameter,  and  which  is  an  inch  hjgh.  All  these  quantities  are 
convertible  into  one  another — -just  as  the  pound  sterling  is  con- 
vertible into  shillings  or  pence,  and  the  ton  weight  is  converti- 
ble into  hundred- weights  and  pounds. 

The  foundation  of  all  mathematical  science  must  be  laid  in  a 
complete  treatise  on  the  science  of  numbers,  and  in  an  accurate 
examination  of  the  different  methods  of  calculation  which  are 
possible  by  their  means.  Now  Arithmetic  treats  of  numbers  in 
particular,  but  the  science  which  treats  of  numbers  in  general 
is  called  Algebra.  In  algebra  numbers  are  expressed  by  letters 
of  the  alphabet,  and  the  advantage  of  the  substitution  is  that 
we  are  enabled  to  pursue  our  investigations  without  being  em- 
barrassed by  the  necessity  of  performing  arithmetical  operations 
at  every  step.  Thus  if  a  given  number  be  represented  by  the 
letter  a,  we  know  that  2  a,  will  represent  twice  that  number, 
and  £  a  the  half  of  that  number,  whatever  the  value  of  a  may 
be.  In  like  manner  if  a  be  taken  from  a,  there  will  be  nothing 
left,  and  this  result  will  equally  hold  whether  a  be  6,  or  7,  or 
1* 


10  ARITHMETIC    OF   THE    STEAM-ENGINE. 

1,000,  or  any  other  number  whatever.  By  the  aid  of  algehra, 
therefore,  we  are  enabled  to  analyse  and  determine  the  abstract 
properties  of  numbers  without  embarrassing  ourselves  with 
arithmetical  details,  and  we  are  also  enabled  to  resolve  many 
questions  that  by  simple  arithmetic  would  either  be  difficult  or 
impossible. 

ADDITION. 

The  first  process  of  arithmetic  is  Addition ;  and  here  the 
first  steps  are  usually  made  by  counting  upon  the  fingers,  as  an 
aid  to  the  perceptions  of  the  total  amount  of  the  quantity  that 
has  to  be  expressed.  For  example,  if  we  hold  up  5  fingers  of 
the  one  hand  and  3  of  the  other,  and  are  asked  how  much  5 
and  3  amount  to,  we  at  once  see  that  the  number  is  8,  as  we 
either  actually  or  mentally  count  the  other  3  fingers  from  5, 
designating  them  as  6,  7,  8 ;  when,  the  whole  fingers  being 
counted,  we  know  that  tbe  total  number  to  be  reckoned  is  8. 
Persons  even  of  considerahte  arithmetical  experience,  will  often 
find  themselves  either  counting  their  fingers  or  pressing  them 
down  successively  on  the  table,  in  order  to  assist  their  memory 
in  performing  addition.  But  the  best  course  is  to  commit  very 
thoroughly  to  memory  an  addition  table,  just  as  the  multiplica- 
tion table  is  now  commonly  committed  to  memory  by  arithmet- 
ical students — as  such  a  table,  if  thoroughly  mastered,  will 
greatly  facilitate  all  subsequent  arithmetical  processes.  A  table 
of  this  kind  is  here  introduced,  and  it  should  be  gone  over  again 
and  again,  until  its  indications  are  as  familiar  to  the  memory  as 
the  letters  of  the  alphabet,  and  until  the  operation  of  addition 
can  be  performed  without  the  necessity  of  mental  effort.  The 
sign  +  placed  between  the  figures  of  the  following  table  is  the 
sign  of  addition  termed  plus,  and  signifies  that  the  numbers  are 
to  be  added  together.  The  table  is  so  plain  as  scarcely  to  re- 
quire explanation.  The  figures  in  the  first  column  are  obtained 
by  adding  together  the  figures  opposite  to  them  in  any  of  the 
other  columns.  Thus  4  and  9  make  13,  as  also  do  5  and  8  or  6 
and  Y. 


METHOD    OF   PERFORMING   ADDITION. 
ADDITION  TABLE. 


11 


2 

1  +  1 

3 

1  +  2 

4 

1  +  3 

2  +  2 

5 

1+4 

2  +  3 

6 

1  +  5 

2  +  4 

3  +  3 

7 

1  +  6 

2  +  5 

3  +  4 

8 

1  +  7 

2  +  6 

3  +  5 

4+4 

9 

1  +  8 

2  +  7 

3  +  6 

4  +  5 

10 

1  +  9 

2  +  8 

3  +  7- 

4+6      5+5 

11 

2  +  9. 

3  +  8 

4  +  7 

5  +  6 

12 

3  +  9 

4  +  8 

5  +  7 

6  +  6 

13 

4  +  9 

6  +  8 

°47 

• 

14 

5  +  9 

6  +  8 

>7  +  7 

15 

6  +  9 

7  +  8 

16 

7  +  9 

8  +  81 

17 

8  +  9 

18 

9  +  9 

GENERAL    EXPLANATION    OF    THE   METHOD   OP   PERFORMING 
ADDITION. 

Write  the  numbers  to  be  added  under  one  another  in  such 
manner  that  the  units  of  all  the  subsequent  lines  of  figures  shall 
stand  vertically  under  the  units  of  the  first  line — the  tens  under 
the  tens,  the  hundreds  under  the  hundreds,  and  so  on.  Then 
add  together  the  figures  found  in  the  units  column.  If  their 
sum  be  expressed  by  a  single  figure,  -write  the  figure  under  the 
units  column,  and  commence  the  same  process  with  the  tens 


12  ARITHMETIC    OF   THE    STEAM-ENGINE. 

column.  But  if  the  sum  of  the  figures  in  the  units  column  he 
greater  than  9,  it  must  in  that  case  he  expressed  in  more  than 
one  figure,  and  in  such  event  write  the  last  figure  only  under  the 
units  column,  and  carry  to  the  column  of  tens  as  many  units  as 
are  expressed  by  the  remaining  figure  or  figures.  Proceed  in 
the  same  manner  with  the  column  of  tens,  and  so  with  all  the 
other  columns.  When  the  column  of  the  highest  order,  which 
is  always  the  first  on  the  left,  has  been  added,  including  the 
number  carried  from  the  column  last  added  up,  then  if  the  sum 
be  expressed  by  a  single  figure,  place  that  figure  under  the  col- 
umn. But  if  it  be  expressed  in  more  figures  than  one,  write 
those  figures  in  their  proper  order,  the  last  under  the  column 
and  the  others  preceding  it. 


Examples. 

Add  togetner  1,904,  9,899,  5,467,  and  2,708.    The  numbers 
are  to  be  arranged  as  follows^ 

1904         Here,  beginning  4*  the  right-hand  column,  we  say  8 
9899    and  7  are  15,  and  9  are  §4,  and  4  are  28.    We  write  the 
2jQo    8  unddr  the  column  of  units,  and  carry  the  2  tens  to  the 
•       next  column  of  tens.    Adding  up  this  column,  we  have 
19,978   the  2  carried  from  the  last  column  added  to  6,  which 
make  8,  and  9  are  17.    Here  we  write  down  the  7  and 
carry  the  1  over  to  the  next  column.    In  the  third  column  we 
have  1  carried  from  the  last  column  added  to  7,  which  makes  8, 
and  4  are  12,  and  8  are  20,  and  9  are  29.    Here  we  write  down 
the  9  and  carry  the  2  to  the  next  column.    In  the  fourth  col- 
umn we  have  the  2  carried  from  the  last  column,  which  added 
to  2  makes  4,  and  5  are  9,  and  9  are  18,  and  1  are  19,  which 
sum  of  19  we  write  at  the  foot  of  the  column,  the  9  under  the 
other  figures  and  the  1  preceding  it.    The  total  sum  of  these 
several  numbers  therefore,  when  added  together,  is  nineteen 
thousand  nine  hundred  and  seventy-eight. 
Add  together  the  following  numbers : — 


USE   OF   COMMAS   IN  NOTATION' — SUBTKACTION.          13 

2808  1467  2708  5794 

1407  5988  5467  9969 

9969  2829  9899  1407 

5794  9694  1904  2808 


19,978  19,978  19,978  19,978 


It  is  usual,  for  facility  of  reading  the  figures,  to  divide  them, 
when  they  amount  to  any  considerable  number,  into  groups  of 
three  each,  by  means  of  a  comma  interposed.  But  the  comma 
in  no  way  affects  the  value  of  the  quantity,  but  is  merely  used 
to  save  the  trouble  of  counting  the  figures  to  make  sure  whether 
it  is  thousands,  hundreds  of  thousands,  or  what  other  order  of 
figures  is  intended  to  be  expressed.  Thus  with  the  aid  of  the 
comma  we  see  at  once  that  the  number  19,000  is  nineteen  thou- 
sand, or  that  the  number  190,000  is  one  hundred  and  ninety 
thousand,  or  that  the  number  1,900,000  is  one  million  nine  hun- 
dred thousand;  whereas,  without  the  aid  of  the  commas,  we 
should  have  to  count  the  figures  to  make  sure  of  the  real  value 
of  the  expression.  The  comma,  therefore,  has  no  such  signifi- 
cance as  the  decimal  point,  and  the  number  may  be  written 
with  or  without  the  comma  at  pleasure ;  but  if  written  without 
it  there  will  be  more  difficulty  in  reading  the  number,  just  as  it 
would  be  more  difficult  to  read  a  book  if  the  stops  were  left  out. 

SUBTRACTION. 

Subtraction  is  the  reverse  of  addition.  If  we  have  a  bag 
containing  20  shillings,  and  if  we  add  thereto  5  shillings,  15 
shillings,  and  10  shillings,  we  can  easily  tell  by  the  operation  of 
addition  that  we  must  have  50  shillings  in  the  bag.  If,  how- 
ever, we  now  withdraw  the  5  shillings,  the  15  shillings,  and  the 
10  shillings,  or,  in  all,  if  we  withdraw  30  shillings,  we  shall,  of 
course,  have  the  original  20  shillings  left ;  and  the  operation  of 
subtraction  is  intended  to  tell  us,  when  we  withdraw  a  less 
number  from  a  greater,  how  much  of  the  greater  number  we 
shall  have  left.  As  addition  is  signified  by  the  sign  +  or  plus, 


14  AEITHMETIG   OF   THE   STEAM-ENGINE. 

so  subtraction  is  signified  by  the  sign  —  or  minus ;  and  two 
short  parallel  lines  =  are  employed  as  a  substitute  for  the  words 
equal  to.  As  the  expression,  therefore,  5  +  3  means  5  increased 
by  3,  or  8;  so  the  expression  5 — 3  means  5  diminished  by  3,  or 
2.  This  in  common  arithmetical  notation  would  be  written 
5  +  3  =  8  and  5  —  3  =  2. 

"When  we  have  a  number  of  quantities  to  subtract  from  a 
greater  quantity,  the  usual  course  is  to  add  together  first  all  the 
quantities  to  be  subtracted,  in  order  that  the  subtraction  may 
be  performed  at  a  single  operation.  Thus  in  the  case  of  the  bag 
containing  50  shillings,  from  which  we  successively  withdraw 
5  shillings,  15  shillings,  and  10  shillings,  we  first  add  together 
the  5  shillings,  the  15  shillings,  and  the  10  shillings,  so  as  to 
have  in  one  sum  the  whole  quantity  to  be  subtracted,  and  then 
we  can  suppose  the  operation  to  be  performed  at  a  single  step, 
as,  the  subtraction  having  been  performed  at  different  times, 
will  not  affect  the  amount  of  the  sum  subtracted  or  the  sum 
left.  Thus  50  —  30  =  20 ;  or  if  we  take  the  successive  stages, 
we  have  50  -  5  =  45,  and  45  -  15  =  30,  and  30  —  10  =  20, 
which  is  the  same  result  as  before. 

GENERAL  EXPLANATION   OF   THE  METHOD   OF  PERFORMING  SUB- 
TRACTION. 

"Write  the  less  number  under  the  greater  in  such  manner 
that  the  units  of  the  second  line  of  figures  shall  stand  vertically 
under  the  units  of  the  first  line — the  tens  under  the  tens,  the 
hundreds  under  the  hundreds,  and  so  on,  as  in  addition.  Draw 
a  straight  line  beneath  the  lower  line  of  figures,  and  subtract 
the  units  of  the  lower  line  of  figures  from  the  units  of  the  up- 
per line,  and  place  the  remainder  vertically  under  the  units  col- 
umn and  beneath  the  straight  line  which  has  been  drawn.  Sub- 
tract the  tens  from  the  tens  in  like  manner,  the  hundreds  from 
the  hundreds,  and  so  on  until  the  whole  is  completed;  and 
where  there  is  no  figure  to  be  subtracted,  the  figure  of  the  up- 
per line  will  appear  in  the  answer  without  diminution,  as  ap 
pears  in  following  examples : 


METHOD   OP  PERFORMING   SUBTRACTION.  15 

1864  Original  number  1864  Original  number 

64  Number  to  be  subtracted  32  Number  to  be  subtracted 

1800  Remainder  1832  Remainder 


From       7854  From       89764384  From       785068473894 

Subtract  6532  Subtract  41341073  Subtract  510054103784 


Answer   1322  Answer  48423311  Answer    275014370110 


In  these  examples  all  the  figures  of  the  second  line  are  less 
than  those  of  the  first  line,  and  we  at  once  see  what  the  re- 
mainder at  each  step  will  be  by  considering  what  sum  we  must 
add  to  the  less  number  to  make  it  equal  to  the  greater.  Thus 
in  subtracting  6532  from  7854,  we  see  that  we  must  add  2  to 
the  2  of  the  lower  line  to  make  the  4  appearing  in  the  upper ; 
and  we  must  add  2  to  the  3  appearing  in  the  lower  line  to  make 
the  6  appearing  in  the  upper.  In  cases ,  however,  in  which 
some  of  the  figures  of  the  lower  line  are  larger  than  those  ex- 
isting in  the  upper,  we  must  borrow  a  unit  from  the  preceding 
column,  which  will  count  as  ten  in  the  column  into  which  it  is 
imported,  and  this  unit  so  borrowed  will  be  added  to  the  sum 
to  be  subtracted  when  that  preceding  column  comes  to  be  dealt 
with.  Thus  in  the  groups  of  stones  used  by  the  natives  of  Mad- 
agascar— if  we  have  6  groups  of  10  stones  in  each  and  7  stones 
over,  and  if  we  want  to  withdraw  8  stones  from  the  number,  it 
is  clear  that,  as  the  7  stones  not  arranged  in  groups  will  not 
suffice  to  supply  the  8  stones  we  have  to  furnish,  we  must  break 
up  one  of  the  groups  of  10  to  enable  the  8  stones  to  be  surren- 
dered. We  shall  then  have  only  5  groups,  but  with  the  7  stones 
we  had  before  we  can  supply  the  8  by  taking  only  one  stone 
from  one  of  the  groups,  leaving  9  stones  in  it,  so  that,  after  tak- 
ing away  the  8  stones,  we  shall  have  5  groups  of  ten  each  and 
9  stones  left.  This  is  expressed  arithmetically  as  follows : 

67  Here  we  say  we  cannot  subtract  8  from  7,  so  that  we 
1  must  borrow  1  from  the  previous  column,  which,  when 

59    imported  into  the  column  of  units,  will  be  10;  and  we 

=    therefore  say  8  taken  from  17  leaves  9,  which  9  we  place 


16  ARITHMETIC   OF   THE   STEAM-ENGINE. 

in  the  remainder.  But  as  we  have  taken  one  of  the  groups 
from  the  preceding  column,  we  have  to  deduct  that  from  the 
six  groups  remaining,  and  we  therefore  say  1  from  6  leaves  5. 
So,  in  like  manner,  if  we  had  to  take  29  shillings  from  42  shil- 
lings, as  we  cannot  take  9  from  2,  we  take  9  from  12,  borrow- 
ing as  before  a  unit  from  the  preceding  column.  But  as  we 
have  afterwards  to  return  this  unit,  we  do  not  say  2  from  4,  but 
3  from  4,  which  leaves  1 ;  or,  in  other  words,  29  taken  from  42 
leaves  13,  as  we  can  easily  see  must  be  the  case,  as  13  added  to 
29  make  42.  To  prove  the  accuracy  of  an  answer  in  subtrac- 
tion, it  is  only  necessary  to  add  together  the  two  lower  lines, 
which  will  produce  the  top  one. 

Examples. 

From 1864  From 1864 

Subtract 14  Subtract...       97 

Remainder     1850  Remainder      1767 


From 1864  From 1864 

Subtract 975  Subtract. ...  1796 


Remainder       889  Remainder         68 


It  will  be  seen  that,  by  adding  together  the  last  two  lines  of 

figures  in  each  of  these  examples,  we  obtain  the  first  line. 

jt 

MULTIPLICATION. 

Multiplication  is  a  process  of  arithmetic  for  obtaining  the 
sum  total  of  a  quantity  that  is  repeated  any  given  number  of 
times,  and  is  virtually  an  abbreviated  species  of  addition.  If, 
for  example,  we  have  6  heaps  of  stones,  with  1,728  stones  in 
each  heap,  we  might  ascertain  the  total  number  of  stones  in  the 
six  heaps  by  writing  the  1,728  six  times  in  successive  lines,  and 
adding  up  the  sum  by  the  method  of  procedure  already  ex- 
plained under  the  head  of  Addition.  But  it  is  clear  that  this 
would  be  a  very  tedious  process  in  cases  in  which  the  number 


MULTIPLICATION   A   SPECIES    OF    ADDITION.  17 

of  heaps  was  great,  and  multiplication  is  an  expedient  for  ascer- 
taining the  total  quantity  by  a  much  less  elaborate  method  of 
procedure. 

All  numbers  whatever  it  is  clear  may  be  formed  by  the  addi- 
tion of  units.  The  consecutive  numbers  1,  2,  3,  4,  5,  &c.,  may 
be  derived  as  follows : 

1  =  1 

1  +  1=2 

1+1+1=3 

1+1+1+1=4 

1+1+1+1+1=5 

There  are  certain  numbers  which  are  formed  by  the  contin- 
ued addition  of  other  numbers  than  1 ;  and  the  numbers  which 
are  formed  by  the  continued  addition  of  2  may  be  shown  as  fol- 
lows: 

2=2 

2  +  2=4 

2+2+2=6 

2+2+2+2=8 

2  +  2  +  2  +  2  +  2=10. 

In  like  manner,  the  numbers  shown  by  the  successive  addi- 
tions of  3  and  4  may  be  thus  represented  : — 

3=3  4=4 

3+3=6  4+4=8  ^ 

3+3+3=9  4+4+4=12 

3+3+3+3—12  4+4+4+4=16 

8+3+3+3+3=15  4+4+4+4+4=20 

Thus  it  will  be  seen  that  in  the  series  of  numbers  proceeding 
upwards  from  1,  some  can  only  be  formed  by  the  continued  ad- 
dition of  1,  while  others  may  be  formed  by  the  continued  addi- 
tion of  2,  3,  or  some  higher  number.  The  numbers  3,  5,  and  7 
cannot  be  produced  by  the  continued  addition  of  any  other 
number  than  1,  while  the  intermediate  numbers  4  and  6  may  be 
formed,  the  first  by  the  addition  of  2,  and  the  second  by  the  con- 
tinued addition  of  2  or  3. 


18  AKITHMETIC    OF   THE    STEAM-ENGINE. 

Those  numbers  which,  cannot  be  formed  by  the  continued 
addition  of  any  other  number  than  1  are  termed  p rime  numbers, 
The  numbers  3,  5,  7,  11,  13,  17,  &c.,  are  prime  numbers.  All 
other  numbers  are  termed  multiple  numbers  ;  and  they  are  said 
to  be  multiples  of  those  lesser  numbers  by  the  continued  addi- 
tion of  which  they  may  be  formed.  Thus  6  is  a  multiple  of  2, 
because  it  may  be  formed  by  the  continued  addition  of  2.  But 
it  is  also  a  multiple  of  3,  because  it  may  be  formed  by  the  con- 
tinued addition  of  3.  In  like  manner  12  is  a  multiple  of  2,  3,  4, 
and  6. 

In  the  ascending  series  of  numbers,  1,  2,  3,  4,  5,  &c.,  it  will 
be  obvious  that  each  alternate  number  is  a  multiple  of  2.  Such 
numbers  are  called  even  numbers,  and  the  intermediate  numbers 
are  called  odd  numbers.  Thus  2,  4,  6,  8,  10,  &c.,  are  even  num- 
bers, and  1,  3,  5,  7,  9,  &c.,  are  odd  numbers. 

As  every  even  number  is  a  multiple  of  2,  it  is  clear  that  no 
even  number  except  2  itself  can  be  a  prime  number,  and  every 
prune  number  except  2  itself  must  be  an  odd  number.  It  by  no 
means  follows,  however,  that  every  odd  number  must  be  prime, 
and  it  is  clear  indeed  that  9  is  a  multiple  of  3,  15  of  3  and  of  5, 
and  so  of  other  odd  numbers,  which  cannot,  therefore,  be  prime 
numbers. 

If  we  take  a  strip  of  paper  an  inch  broad  and  12  inches  long, 
like  a  strip  of  postage  stamps,  it  is  clear  that  this  strip  will  con- 
tain 12  square  inches ;  and  if  we  take  three  such  strips  placed 
side  by  side,  they  will  manifestly  have  a  collective  surface  of  36 
square  inches.  Nor  will  the  result  be  different  in  whatever  way 
we  reckon  the  squares ;  and  12  multiplied  by  3  will  give  just  the 
same  number  as  3  multiplied  by  12.  In  like  manner,  7  multi- 
plied by  5  is  the  same  as  5  multiplied  by  7,  and  so  of  all  other 
numbers. 

In  order  to  be  able  to  perform  the  operations  of  multiplication 
with  ease  and  expedition,  it  is  necessary  to  commit  to  memory 
the  product  of  the  multiplications  of  numbers  from  1  to  9 ;  and 
to  enable  this  to  be  conveniently  done,  a  table  of  these  primary 
multiplications,  called  the  Multiplication  Table,  forms  part  of 
the  course  of  arithmetical  instruction  given  at  schools,  where, 


THE    MULTIPLICATION   TABLE. 


19 


however,  the  tables  used  commonly  carry  the  multiplications  up 
to  12  times  12.  A  table  containing  all  the  multiplications  neces- 
sary to  be  remembered  is  given  below ;  and  it  is  very  material  to 
the  subsequent  ease  of  all  arithmetical  processes,  that  this  table 
should  be  thoroughly  learned  by  heart,  so  as  to  obviate  the  hesi- 
tation and  inaccuracy  that  must  otherwise  ensue. 

MULTIPLICATION  TABLE. 


2 

3 

4 

5 

6 

I 

8   9 

9 

18 

27 

36 

45 

54 

63 

72   81 

8 

16 

24 

32 

40 

48 

56 

64 

7 

14 

21 

28 

35 

42 

49 

6 

12 

18 

24 

30 

36 

5 

10 

15 

20 

25 

4 

8 

12 

16 

3 

6 

9 

2 

4 

To  find  the  product  of  two  numbers  by  this  table,  we  must 
look  for  the  greater  number  in  the  first  upright  column  on  the 
left,  and  for  the  lesser  number  in  the  highest  cross  row.  The 
product  of  the  two  numbers  will  be  found  in  the  same  cross  row 
with  the  greater  number,  and  hi  the  same  upright  column  with 
the  lesser  number.  Thus  6  times  3  are  18,  6  times  4  are  24,  and 
6  times  4  are  20.  If  we  find  the  number  6  in  the  first  column 
and  pass  our  finger  along  the  same  line  until  we  come  vertically 
under  the  3  in  the  top  line,  we  find  the  number  18,  which  is  the 
product  required.  By  the  same  process  we  find  the  numbers  24 
and  20. 

Having  committed  the  multiplication  table  to  memory,  we 
are  in  a  condition  for  performing  any  multiplication  of  common 


20  AKITHIUETIC   OP   THE   STEAM-ENGINE. 

numbers  without  difficulty.     If,  for  example,  we  wish  to  multiply 
1,728  by  2,  we  write  the  2  under  the  8  and  draw  a  line  thus  :  — 

are  16.    We  write  down  the  6 


1728 

2    and  carry  the  1,  which  belongs  to  the  order  of  tens  next 

3456  above,  to  that  order.  Twice  2  are  4,  and  the  1  carried 
-  from  the  16  of  the  last  multiplication  make  5.  The  num- 
ber 5  being  less  than  10,  there  is  no  figure  to  carry  in  this  case. 
"We  therefore  say  twice  7  are  14,  where  again  we  write  4  and 
carry  1,  and  twice  1  are  2,  and  1  carried  over  from  the  last  mul- 
tiplication make  3. 

It  is  clear  that  the  number  1,728  is  made  up  of  the  numbers 
1,000,  700,  20,  and  8,  and  the  result  of  the  multiplication  would 
not  be  altered  if  we  were  to  multiply  these  quantities  separately 
and  add  them  together.  A  Saint  Andrew's  cross  or  x  is  the 
sign  of  multiplication  ;  and 

1000  x  2=2000 

700x2=1400 

20x2=     40 

8x2=     16 

3456 


Here,  then,  we  see  we  have  precisely  the  same  result  as  in  the 
former  case.  But  the  first  expedient  is  the  simpler,  and  is  there- 
fore commonly  used.  We  shall  also  obtain  the  same  result  by 
adding  1,728  to  1,728,  thus  :— 

j^g  In  this  particular  case  it  is  as  easy  to  add  the  number 
1728  to  itself  as  to  multiply  by  2.  But  when  the  multiplica- 
3456  tion  proceeds  to  6,  8,  or  any  greater  number  of  times,  it 
:  would  be  very  inconvenient  to  have  to  add  the  number 
to  itself  6  or  8  times,  and  it  is  much  easier  to  proceed  by  the 
common  method  of  multiplication  here  explained.  The  number 
we  multiply  with  is  called  the  multiplier,  and  the  number  we 
multiply  is  called  the  multiplicand,  while  the  number  resulting 
from  the  multiplication  is  called  the  product.  In  the  above  ex- 
ample 2  is  the  multiplier,  1,728  the  multiplicand,  and  3,456  the 
product. 


MULTIPLIERS  CONTAINING  CIPHERS.  21 

If  the  multiplier  consists  of  two  figures  instead  of  one,  the 
same  mode  of  procedure  is  pursued,  except  that  the  whole  of  the 
figures  resulting  from  the  multiplication  of  the  higher  of  the  two 
figures  is  shifted  one  place  to  the  left.  Thus,  if  the  number 
1,Y28  has  to  be  multiplied  by  22,  the  mode  of  procedure  is  as 
follows : — 

Here  the  arithmetical  process  of  multiplication  is 
o2    precisely  the  same  with  each  of  the  two  figures,  only 
that  in  the  case  of  the  second  multiplication  the  result- 
j   !6    ing  number  is  set  one  place  more  to  the  left ;  and  the 
two  lines  of  partial  products  are  then  added  together 


38,016  for  tbe  answer.  It  is,  therefore,  a  rule  in  all  multipli- 
cations where  the  multiplier  consists  of  more  figures 
than  one,  that  the  first  figure  of  the  product  shall  be  set  under 
that  particular  figure  of  the  multiplier  with  which  that  particular 
line  of  multiplication  is  performed.  If  instead  of  22  the  multi- 
plier had  been  222,  then  the  operation  would  have  been  as 
follows : — 

Here,  it  will  be  observed,  the  same  partial  product 
-go    is  repeated  in  every  case,  but  set  one  place  more  to  the 
left ;  and  the  several  lines  of  partial  products  are  then 
added  up  for  the  total  product  of  the  multiplication. 
3456  In  cases  where  one  of  the  figures  of  the  multiplier 

is  a  cipher,  the  only  effect  is  to  shift  the  figures  over  to 
'  '  ' '  the  left  one  place,  and  which  may  be  done  by  adding  a 
cipher  to  the  product  if  the  cipher  forms  the  last  figure 
of  the  multiplier.  Thus,  1,728  multiplied  by  20,  is  34,560,  mul- 
tiplied by  200  is  345,600,  and  multiplied  by  2,000  is  3,456,000. 
If  the  cipher  comes  in  the  middle  of  the  multiplier,  as  in  multi- 
plying by  202,  we  proceed  as  follows : — 

Here  we  pass  over  the  cipher  altogether,  except  that 
202    we  begin  the  succeeding  line  of  multiplication  one  place 
more  to  the  left  than  we  should  have  done  if  the  cipher 

* 

had  not  been  present ;  or,  in  other  words,  we  begin  the 
line  pertaining  to  the  next  figure  of  the  multiplier  un- 
'    der  that  figure,  just  as  would  be  done  if  any  other 
figure  than  a  cipher  intervened.      Indeed  we  might 


22  AEITHMETIC   OP  THE   STEAM-ENGINE. 

write  a  line  of  ciphers  as  resulting  from  multiplication  by  a 
cipher ;  but  as  this  line  could  not  affect  the  value  of  the  sum 
total,  it  is  left  out  altogether.  In  multiplying  numbers  termi- 
nating with  ciphers,  or  in  multiplying  with  numbers  terminating 
with  ciphers,  the  mode  of  procedure  is  to  perform  the  multipli- 
cation as  if  there  were  no  ciphers,  and  then  to  annex  as  many 
ciphers  to  the  product  as  there  are  ciphers  in  the  multiplier  and 
multiplicand  together.  Thus  65,000  multiplied  by  3,300  is 
treated  as  if  65  had  to  be  multiplied  by  33,  and  then  five  ciphers 
are  added  to  the  product  to  give  the  correct  answer. 

GENERAL   EXPLANATION   OF   THE  METHOD   OF   PEEFOBMINa 
MULTIPLICATION. 

The  foregoing  explanations  of  the  method  of  performing  the 
multiplication  of  numbers  will  probably  suffice  to  enable  all  or- 
dinary questions  in  multiplication  to  be  readily  performed.  But 
for  the  sake  of  clearness,  it  may  be  useful  to  recapitulate  the 
several  steps  of  the  process. 

Place  the  multiplier  under  the  multiplicand,  as  in  addition. 
Multiply  the  multiplicand  separately  by  each  significant  figure 
of  the  multiplier,  by  which  we  shall  obtain  as  many  partial  prod- 
ucts as  there  are  significant  figures  in  the  multiplier.  "Write 
these  products  under  one  another,  so  that  the  last  figure  of  each 
shall  be  under  that  figure  of  the  multiplier  by  which  it  has  been 
produced.  Add  the  partial  products  thus  obtained,  and  their 
sum  will  be  the  total  product. 

It  will  often  facilitate  arithmetical  calculations  if  we  have 
committed  to  memory  the  products  of  numbers  larger  than  those 
found  in  the  common  multiplication  tables,  and  it  is  very  impor- 
tant that  these  elementary  multiples  should  be  accurately  and 
promptly  recollected.  In  the  following  table  the  products  of 
numbers  are  given  as  high  as  20  times  20 : 


MULTIPLICATION  TABLE  EXTENDING  TO  20  TIMES  20.    23 


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24  ARITHMETIC   OF  THE   STEAM-ENGINE. 

DIVISION. 

When  a  number  has  to  be  separated  into  two,  three,  or  any 
other  number  of  equal  parts,  it  is  done  by  means  of  Division, 
which  enables  us  to  determine  the  magnitude  of  one  of  those 
parts.  If,  for  example,  we  wish  to  divide  12  inches  into  four 
equal  parts,  the  length  of  each  of  those  parts  will  be  3  inches. 
If  we  wish  to  divide  it  into  three  equal  parts,  the  length  of  each 
of  the  parts  will  be  4  inches ;  or  if  we  wish  to  divide  it  into  two 
equal  parts,  the  length  of  each  part  will  be  6  inches. 

The  number  which  is  to  be  decomposed  or  divided  is  calledj 
the  dividend,  the  number  of  equal  parts  into  which  the  number 
sought  to  be  divided  is  called  the  divisor,  and  the  magnitude  of 
one  of  those  parts  obtained  from  the  division  is  called  the  quo- 
tient. Thus  in  dividing  12  by  3, 

12  is  the  dividend, 

3  is  the  divisor, 

4  is  the  quotient. 

It  follows  from  this  explanation  of  the  process  of  division, 
that  if  we  divide  a  number  into  two  equal  parts,  one  of  those 
parts  taken  twice  will  reproduce  the  original  number ;  or  if  we 
divide  it  into  three  equal  parts,  one  of  those  parts  taken  three 
times  will  reproduce  the  original  number.  In  all  cases,  indeed, 
the  quotient  multiplied  by  the  divisor  will  produce  the  dividend. 
Hence  division  is  said  to  be  a  rule  which  teaches  us  to  find  a 
number  which,  multiplied  by  the  divisor,  will  reproduce  the 
dividend.  For  example,  if  35  has  to  be  divided  by  5,  we  seek 
for  a  number  which,  multiplied  by  5,  will  produce  35.  This 
number  is  7,  since  5  times  7  is  35.  The  manner  of  expression 
employed  in  this  division  is  5  in  35  goes  7  times,  and  5  tunes  7 
makes  35.  The  dividend,  therefore,  may  be  considered  as  a  prod- 
uct, of  which  one  of  the  factors  is  the  divisor  and  the  other  the 
quotient.  Thus,  supposing  we  have  63  to  divide  by  7,  we  en- 
deavour to  find  such  a  product  that,  taking  7  for  one  of  its  fac- 
tors, the  other  factor  multiplied  by  this  shall  produce  exactly 
63.  Now  7  x  9  is  such  a  product,  and,  conseauently,  9  is  the 
quotient  obtained  when  we  divide  63  by  7. 


NATURE   OF   DIVISION.  25 

In  the  same  sense  in  which  multiplication  above  unity  may 
be  looked  upon  as  a  continued  addition,  so  may  division  be  looked 
upon  as  a  continued  subtraction.  Thus  as  7x9  =  74-7  +  7+7  + 
7+7+7+7+r,  so  also  63  -~  9  =  63-7-7-7-7-7- 7-7-7. 
This  may  easily  be  seen  by  performing  the  operation  of  addition 
or  subtraction.  Thus  7  and  7  are  14,  and  14  and  7  are  21,  and  21 
and  7  are  28,  and  28  and  7  are  35,  and  35  and  7  are  42,  and  42 
and  7  are  49,  and  49  and  7  are  56,  and  56  and  7  are  63.  So  in 
like  manner  63  less  7  are  56,  and  56  less  7  are  49,  and  49  less  7 
are  42,  and  42  less  7  are  35,  and  35  less  7  are  28,  and  28  less  7 
are  21,  and  21  less  7  are  14,  and  14  less  7  are  7,  and  7  less  7 
is  0. 

"We  have  seen  that  when  we  divide  12  inches  by  4,  we  ob- 
tain 3  inches  as  the  quotient.  But  if  we  divide  13  inches  by  4 
we  shall  have  4  parts  of  3  inches  each  and  1  inch  over,  and  if 
this  inch  be  also  divided  into  4  equal  parts,  each  of  these  parts 
will  be  one  quarter  of  an  inch.  Hence  13  inches  divided  by  4 
gives  3i  inches.  So  if  we  divide  63  feet  into  lengths  of  7  feet 
each  we  shall  have  exactly  9  of  such  lengths.  But  if  we  divide 
64  feet  into  lengths  of  7  feet  each,  we  shall,  after  having  per- 
formed the  division,  have  1  foot  over.  This  foot  is  obviously 
just  one  sixty- third  of  the  total  length ;  and  if  we  wish  to  dis- 
tribute this  residual  foot  equally  among  the  whole  of  the  other 
divisions,  we  must  either  divide  it  into  9  equal  parts,  and  add  1 
of  these  parts  to  each  division,  or  we  must  divide  it  into  63  equal 
parts,  and  add  1  of  these  parts  to  each  foot,  or  7  of  them  to  each 
division.  It  follows  that  64  divided  by  9  is  equal  to  7g-,  or  to 
Tffj,  which  is  the  same  thing.  So  in  dividing  a  plank  50  feet 
long  into  lengths  of  4  feet  each,  we  shall  have  12  such  lengths  in 
the  length  of  the  plank,  and  we  shall  have  2  feet  over.  If  we 
wish  to  distribute  these  2  feet  equally  among  the  12  divisions,  so 
that  no  part  of  the  plank  may  be  cut  to  waste,  then  we  must  in- 
crease the  length  of  each  foot  one  forty-eighth  part  of  2  feet,  or 
we  must  increase  the  length  of  each  division  one-twelfth  part 
of  2  feet,  or  two-twelfth  parts  of  1  foot.  Now,  as  the  foot  con- 
sists of  12  inches,  two-twelfth  parts  are  equal  to  2  inches.  More- 
over, as  a  twenty-fourth  part  of  a  foot  is  equal  to  half  an  inch, 
2 


« 
i 


26  ARITHMETIC    OF   THE    STEAM-ENGINE. 

and  a  forty-eighth  part  of  a  foot  is  equal  to  a  quarter  of  an  inch, 
it  follows  that  8  forty-eighth  parts  are  equivalent  to  8  quarters 
of  an  inch,  or  to  2  inches,  as  before.  Each  division  of  the  plank 
of  50  feet,  therefore,  must  he  4  feet  2  inches  long,  in  order  that 
it  may  he  cut  without  waste  into  12  equal  lengths. 

If  we  have  a  number  50  which  we  wish  to  divide  by  another 
number  12,  then  we  write  the  number  as  follows : — 

"We  say  the  twelves  in  50,  4  times  and  2  over,  which  two- 

12)60      twelfths  is  written  as  a  vulgar  fraction,  and  forms 

—      part  of  the  quotient.    But  if  we  wish  the  answer  to 

* ii  be  in  decimal  fractions,  we  place  a  decimal  point  after 

the  50,  and  add  thereto  any  number  of  ciphers,  continuing  the 

division  in  precisely  the  same  manner  as  if  the  number  were  not 

a  fraction  at  all.    Thus — 

12)50-00000  Here  we  say,  as  before,  the  twelves  in  50,  4 

times  and  2  over,  which  2  we  carry  to  the  next 

4-16666,  &c.  succeeding  place  of  figures,  and  say  the  twelves 
in  20  once  and  8  over,  the  twelves  in  80  6  times  and  8  over,  the 
twelves  in  80  6  times  and  8  over,  and  so  on  to  infinity.  "We 
thus  see  not  merely  that  the  fraction  f^ths  or  -Jth,  called  the 
remainder,  is  left  over  when  we  divide  50  by  12,  but  that  this 
fraction  may  be  expressed  decimally  under  the  form  of  the  infi- 
nite series  of  numbers  "16666,  &c.,  which  numbers,  if  carried  on 
for  ever,  will  be  continually  coming  nearer  to  the  quantity  ^th, 
but  will  never  be  absolutely  equal  to  it,  though  sufficiently  near 
thereto  to  answer  all  the  purposes  of  practical  computation. 

A  very  little  consideration  will  suffice  to  show  us  the  reason 
of  the  process  in  division  in  which  we  carry  the  residual  num- 
ber to  the  next  place  of  figures  immediately  succeeding.  Thus, 
if  we  have  to  divide  the  number  963  by  3,  we  may,  if  we  please, 
perform  the  operation  by  dividing  the  whole  number  into  900, 
60,  and  3,  and  dividing  them  separately.  Now  the  third  of  900 
is  obviously  300,  the  third  of  60  is  20,  and  the  third  of  3  is  1,  so 
that  the  third  of  the  total  number  of  963  is  321.  If,  however, 
the  number  which  we  had  to  divide  by  3  was  954,  then  in  divid- 
ing the  constituent  numbers  as  before,  we  should  have  the  third 
of  900  which  is  300,  the  third  of  50  which  is  16,  leaving  2  over, 


LONG    AND  SHORT    DIVISION.                               27 

•which  2  has  to  be  added  to  the  4  not  yet  divided,  making  it  up 

to  6 ;  and  the  third  of  6  is  2.  These  numbers  may  be  written  as 
follows : — 

900  divided  by  3=300  900  divided  by  3  =  300 

60  divided  by  3=  20  50  divided  by  3=   16 

3  divided  by  3=     1  6  divided  by  3=     2 

321  318 


By  the  ordinary  method  of  division,  the  quantity  would  bo 
written  thus : — 

Divisor  3)963  Dividend  Divisor  3)954  Dividend 

821  Quotient.  318  Quotient. 

Here,  in  the  first  example,  we  say  the  threes  in  9,  three  times, 
which  3  we  write  under  the  9 ;  the  threes  in  6  twice,  which  2 
we  write  under  the  6 ;  and  the  threes  in  3  once,  which  1  we 
write  under  the  3.  In  the  second  example  we  say,  as  before, 
the  threes  in  9  three  times ;  but  the  threes  in  5  will  only  go 
once,  leaving  2  as  a  remainder,  which  2  when  imported  into  the 
next  inferior  place  of  figures,  will  count  ten  times  greater,  or  as 
20 ;  and  we  then  say  the  threes  in  24  eight  times,  which  8  we 
write  under  the  4.  It  will  be  recollected  that  as  the  second 
place  of  figures  from  the  right  is  groups  of  tens,  two  of  these 
groups  when  resolved  into  units  must  necessarily  be  20. 

The  method  of  division  here  described  is  that  used  when 
'  any  number  has  to  be  divided  by  another  number  consisting  of 
only  one  figure.  It  is  called  Short  Division.  In  the  case 
quantities  which  have  to  be  divided  by  numbers  consisting  o: 
two  or  more  figures,  this  method  would  not  be  convenient,  and 
another  method  called  Long  Division  is  commonly  employed. 
If,  for  example,  we  had  to  divide  4967398  by  37,  we  may,  no 
doubt,  perform  the  question  by  the  method  of  short  division. 
But  the  remainders,  when  there  are  several  figures  in  the  di- 


28  ARITHMETIC    OF   THE    STEAM-ENGINE. 

visor,  become  so  large  and  perplexing,  that  it  is  much,  hetter  to 
employ  the  method  of  long  division,  which  is  as  follows : 

Dividend 

Divisor  37)4967398(134254  Quotient 
37 

126 
111 

157 

148 

93 

74 

199 
185 

148 
148 


Here  we  first  find  how  many  times  37  are  contained  in  49, 
and  it  is  clear  it  is  contained  only  once.  We  write  therefore  1 
in  the  quotient,  and  multiply  the  divisor  by  it,  placing  the  prod- 
uct under  the  49,  and  we  subtract  the  37"  from  the  49,  which 
shows  that  there  is  a  remainder  of  12.  To  this  remainder  we 
next  bring  down  the  figure  of  the  original  number  which  imme- 
diately succeeds  the  49,  and  which  in  this  case  is  6.  We  then 
consider  how  many  times  37  are  contained  in  126,  and  we  find 
that  it  is  three  times.  We  write  the  3  in  the  quotient,  and 
multiply  the  divisor  by  it,  when  we  find  that  the  product  is  111, 
which  sum  we  subtract  from  the  126,  and  find  that  we  have  a 
remainder  of  15.  To  this  15  we  next  bring  down  the  figure  of 
the  original  sum  succeeding  to  that  which  we  brought  down  be- 
fore, and  which  in  this  case  is  7,  and  we  consider  how  many 
times  37  will  go  in  157.  We  find  that  it  will  go  four  times,  and 
we  write  the  4  in  the  quotient  as  before,  and  proceed  to  multi- 
ply the  divisor  by  it,  and  to  subtract  the  product  148  from  the 
159,  which  will  leave  a  remainder  of  9.  Carrying  on  this  pro- 
cess until  we  have  successively  brought  down  all  the  figures  of 


DIVIDING    BY   THE    FACTORS    OF   A   NUMBER.  29 

the  original  sum  that  had  to  be  divided,  we  find  that  the  quo- 
tient is  134254,  which  number,  if  multiplied  by  37,  will  repro- 
duce the  4967398  with  which  we  set  out.  When  after  perform- 
ing the  division  there  is  found  to  be  a  remainder,  it  may  either 
be  written  as  the  numerator  of  a  vulgar  fraction  in  the  answer, 
the  divisor  being  the  denominator,  or  a  decimal  point  may  be 
introduced  after  the  last  figure,  and  any  desired  number  of  ci- 
phers may  be  added  thereto,  when,  by  continuing  the  division, 
the  remainder  will  be  obtained  in  decimal  fractions. 

The  operation  of  division  is  indicated  by  the  sign  -5-  and  as 
12x12=144,  so  144-f-12=12. 

In  cases  in  which  the  divisor  is  composed  of  two  factors,  it 
is  a  common  practice,  instead  of  employing  the  method  of  long 
division  to  divide  successively  by  the  two  factors  by  the  method 
of  short  division,  which  is  more  rapidly  done.  Thus  if  a  num- 
ber has  to  be  divided  by,  say  36,  the  same  result  will  be  ob- 
tained if  it  is  divided  by  6  and  the  quotient  be  then  again  di- 
vided by  6.  Or,  if  we  have  to  divide  by  42,  we  may  divide  by 
6  and  then  by  7 ;  if  we  have  to  divide  by  63,  we  may  divide  by 
9  and  then  by  7 ;  and  so  of  all  other  numbers  possessing  similar 
factors. 

As,  by  annexing  a  cipher  at  the  end  of  any  number,  we  mul- 
tiply its  amount  by  1Q,  so  by  abstracting  a  cipher  from  the  end 
of  any  number  we  divide  its  amount  by  10.  Thus  2  x  10=20 
and  20x10=200.  So  also  200-5-10=20  and  20-4-10  =  2.  If, 
therefore,  we  have  a  divisor  containing  a  number  of  ciphers,  we 
may  leave  them  out  of  the  account  in  performing  division  :  but 
in  such  case  we  must  count  off  as  decimals  an  equal  number  of 
figures  as  we  have  excluded  of  ciphers.  Thus  l728-5-10=l72'8 
or  1728-^-100=17-28  or  !728-*-1000=r728.  So  444-5-20=22-2 
and  999-5-30=33-3  or  999-5-300=3-33. 

GENEBAL  EXPLANATION   OP   THE  METHOD   OF   PERFOKMINO 
DIVISION. 

Short  Division. — Divide  the  first  figure  of  the  dividend  by 
the  divisor,  and  place  the  quotient  under  the  same  figure  of  the 
dividend.  Prefix  the  remainder  to  the  next  figure  of  the  divi- 


30  ARITHMETIC    OF   THE    STEAM-ENGINE. 

dend  and  divide  the  number  thus  obtained  by  the  divisor.  Place 
the  quotient  under  the  second  figure  of  the  dividend,  and  prefix 
the  remainder  to  the  third  figure  of  the  dividend.  Divide  the 
number  thus  obtained  by  the  divisor,  and  proceed  as  before, 
continuing  this  process  until  you  arrive  at  the  units  place  of  the 
dividend,  when  the  division  will  be  complete. 

Long  Division. — "Write  the  divisor  on  the  left  of  the  divi- 
dend, separated  from  it  by  a  line.  Place  another  line  to  the 
right  of  the  dividend  after  the  units  place  to  separate  the  quo- 
tient from  the  dividend — the  quotient  being  afterwards  written 
on  the  right  of  that  line. 

Count  off  from  the  left  of  the  dividend  or  from  its  highest 
place  as  many  figures  as  there  are  places  in  the  divisor.  If  the 
number  formed  by  these  be  less  than  the  divisor,  then  count  off 
one  more.  Consider  these  figures  as  forming  one  number,  and 
find  how  often  the  divisor  is  contained  in  that  number.  It  will 
always  be  contained  in  it  less  than  ten  times,  and  therefore  the 
quotient  will  always  consist  of  a  single  figure.  Place  this  sin- 
gle figure  as  the  first  figure  of  the  quotient. 

Multiply  the  divisor  by  this  single  figure,  and  place  the  prod- 
uct under  those  figures  of  the  dividend  which  were  taken  off  on 
the  left,  and  subtract  such  product  from  the  number  above  it, 
by  which  we  obtain  the  first  remainder.  This  remainder  must 
always  be  less  than  the  divisor. 

On  the  right  of  the  first  remainder  place  that  figure  of  the 
dividend  which  next  succeeds  those  which  were  cut  off  to  the  left. 
Find  how  often  the  divisor  is.  contained  in  the  number  thus 
formed,  and  place  the  resulting  ^figure  of  the  quotient  next  to 
the  figure  of  the  quotient  already  found.  Multiply  the  divisor 
by  this  figure,  and  proceed  as  before,  until  all  the  succeeding 
figures  of  the  dividend  have  been  brought  down,  when  the  di- 
vision will  be  complete. 

NATUEE   AND   PEOPEETIE9   OF   FEAOTIONS. 

It  has  already  been  stated  that  a  fraction  which  has  the  same 
numerator  and  denominator  is  exactly  equal  to  1,  and  therefore 


NATURE  AND  PROPERTIES  OF  FRACTIONS.      31 

such  a  fraction  is  of  the  same  value  as  an  integer  or  whole  num- 
ber.    For  example,  the  fractions 

i,  f,  -t,  I,  I,  -f,  I,  I,  &c., 

are  all  equal  to  1,  and  are  therefore  equal  to  one  another. 

All  fractions  of  which  the  numerator  is  less  than  the  denom- 
inator have  a  less  value  than  unity  ;  for  if  a  number  be  divided 
by  another  number  greater  than  itself,  the  result  must  be  less 
than  1.  If  we  cut  a  lath  2  feet  long  into  three  equal  lengths, 
one  of  those  lengths  will  certainly  be  shorter  than  a  foot. 
Hence  it  is  evident  that  f  is  less  than  1,  and  for  the  reason  that 
the  numerator  2  is  less  than  the  denominator  3.  If,  on  the  con- 
trary, the  numerator  be  greater  than  the  denominator,  then  it 
will  follow  that  the  fraction  will  be  greater  than  1.  Thus  f  is 
greater  than  1,  for  f  is  equal  to  f  and  J,  and  as  f  is  equal  to  1, 
then  f  will  be  equal  to  1$.  In  the  same  manner  f  is  equal  to  1£, 
|-  to  If,  ^  to  2J,  and  so  on.  In  all  such  cases  it  is  sufficient  to 
divide  the  upper  number  by  the  lower,  and  if  there  is  a  remain- 
der, to  write  it  as  the  numerator  of  the  residual  fraction,  and 
the  divisor  as  the  denominator.  If,  for  example,  the  fraction 
were  ff  ,  we  should  divide  the  43  by  12,  when  we  should  get  3 
as  the  integer  and  ^  as  a  remainder  ;  or,  in  other  words,  we 
should  obtain  the  number  3^-.  Fractions  like  ff  ,  which  have 
the  numerator  greater  than  the  denominator,  are  termed  im- 
proper fractions,  to  distinguish  them  from  fractions  properly  so 
called,  which,  having  the  numerator  less  than  the  denominator, 
are  less  than  unity,  or  an  integer. 

As  we  can  only  understand  what  the  fraction  -^  is  when  we 
know  the  meaning  of  ^  we  may  consider  the  fractions  whose 
numerator  is  unity  as  the  foundation  of  all  others.  Such  are 
the  fractions 


and  it  is  observable  that  these  fractions  go  on  continually  dimin- 
ishing, for  the  more  we  divide  an  integer,  or  the  greater  the 
number  of  parts  into  which  we  distribute  it,  the  less  does  each 
part  become.  Thus  TJ^  is  less  than  ^  ;  -^^  is  less  than  ^  ; 
**  I688  toan  -nfa  ;  and  as  we  increase  the  denominator  of 


32  ARITHMETIC    OF   THE    STEAM-ENGINE. 

the  fraction,  the  less  does  the  value  of  the  fraction  become.  If, 
therefore,  we  suppose  the  denominator  to  be  made  infinitely 
large,  the  fraction  will  become  equal  to  nothing.  To  express 
the  idea  of  infinity,  we  make  use  of  the  symbol  o>,  and  we  may, 
therefore,  say  that  the  fraction  ^-=0.  Now,  we  know  that  if 
we  divide  the  dividend  1  by  the  quotient  -§£,  which  is  equal  to 
nothing,  we  obtain  again  the  divisor  oo .  Hence  we  learn  that 
infinity  is  the  quotient  arising  from  the  division  of  1  by  0.  Thus 
1  divided  by  0  expresses  a  number  infinitely  great.  But  £  is 
certainly  only  the  half  of  f ,  or  the  third  of  §  ;  so  that  it  would 
appear  as  if  one  infinity  may  be  twice  or  three  times  greater 
than  another.  It  will  be  obvious  that  as  the  fractions 

f ,  f,  f,  1,  f ,  "k ,  I,  1,  &c-, 
are  all  equal  to  one  another,  each  of  them  being  in  fact  equal  to 

1,  so  also  the  fractions 

f,  I,  f ,  I,  Y,  ¥,  ¥,  &c., 

are  all  equal  to  one  another,  each  of  them  being  in  fact  equal  to 
2 ;  for  the  numerator  of  each  divided  by  the  denominator  gives 

2.  So  likewise  the  fractions 

f ,  I,  I,  ¥,  ¥,  ¥»  ¥»  &c., 

are  equal  to  one  another,  since  in  fact  each  of  them  is  equal  to  3. 
Now  it  is  clear  that  as  |  is  the  same  as  -^-,  and  as  f  is  the 
same  as  J/,  both  being  equal  to  3,  the  value  of  a  fraction  will 
not  be  changed  if  we  multiply  numerator  and  denominator  by 
the  same  number.  Thus  in  the  case  of  the  fraction  £,  if  we 
multiply  numerator  and  denominator  by  4,  we  shall  have  f , 
which  is  clearly  equal  to  £.  So  also  the  fractions 

i,  f ,  f ,  f,  ^0,  TV,  &,  A,  M,  &c., 
are  equal,  each  of  them  being  equal  to  £.    The  fractions 

t,  f»  I.  A,  A,  A,  A.  A>  A*  t*f  &c., 
are  also  equal,  each  being  equal  to  4;  and  the  fractions 

t,  *,  I,  A>  ii,  if,  M,  H,  i-f »  &o., 
are  also  equal,  each  of  them  being  equal  to  jj. 


REDUCING   FRACTIONS    TO    LOWEST   TERMS.  33 

Now  of  all  the  equivalent  fractions 


the  quantity  -|  is  that  of  which  it  is  easiest  to  form  a  definite 
idea.  It  is  usual,  therefore,  when  we  have  any  such  fraction  as 
£f-  or  Jf,  to  reduce  it  to  its  lowest  terms,  by  dividing  numerator 
and  denominator  by  some  number  that  will  divide  each  without 
a  remainder.  This  division,  it  is  clear,  will  not  affect  the  nu- 
merical value  of  the  fraction  ;  for  if  we  can  multiply  both  nu- 
merator and  denominator  by  the  same  number  without  affecting 
the  value,  so  we  may  divide  both  without  affecting  the  value  ; 
as  by  such  division  we  bring  back  the  fraction  of  which  both 
portions  had  been  multiplied  to  the  original  expression. 

The  number  by  which  the  numerator  and  denominator  of  a 
fraction  may  be  divided  without  leaving  a  remainder  is  called  a 
common  divisor  ;  and  so  long  as  we  can  find  for  the  numerator 
and  denominator  a  common  divisor,  it  is  certain  that  the  frac- 
tion may  be  reduced  to  a  lower  form.  But  if  we  cannot  find 
such  common  divisor,  the  fraction  is  in  its  lowest  form  already. 
Thus  in  the  fraction  -j^,  we  at  once  see  that  both  terms  are  di- 
visible by  2,  and,  performing  this  division,  the  fraction  becomes 
|4  ;  which,  if  again  divided  by  2,  becomes  |-f  ,  and  which  in  like 
manner,  by  another  division  by  2,  becomes  •£$.  This,  it  will  be 
obvious,  cannot  any  longer  be  divided  by  2,  but  it  may  be  by  3, 
when  the  expression  becomes  f  ;  and  as  this  cannot  be  divided 
by  any  other  number  than  1,  it  follows  that  the  fraction  is  now 
in  its  lowest  terms.  Now  2  x  2  x  2  x  3=24,  and  instead  of  the 
successive  divisions  by  2,  by  2,  by  2,  and  by  3,  we  may  divide 
at  once  by  the  product  of  these  quantities,  or  24;  and  dividing 
numerator  and  denominator  of  •£/•$  by  24,  we  have  £  as  before. 

The  property  of  fractions  retaining  the  same  value,  whether 
we  multiply  or  divide  their  numerator  and  denominator  by  the 
same  number,  carries  this  important  consequence  —  that  it  ena- 
bles fractions  to  be  easily  added  or  subtracted,  after  having  first 
brought  them  to  the  same  denomination.  If,  for  example,  we 
had  to  add  together  f  ,  -J,  ^,  and  ^  of  an  inch,  we  could  not  do 
so  easily  unless  we  brought  the  whole  of  these  quantities  to 
2* 


34  ARITHMETIC    OF   THE    STEAM-ENGINE. 

thirty-seconds.  When  so  reduced  the  quantities  will  be  |f  ,  -g^-, 
^-,  /-%,  the  sum  of  which  is  clearly  ff  ,  or,  dividing  numerator 
and  denominator  by  8,  the  expression  becomes  f. 

All  whole  numbers,  it  is  clear,  may  be  expressed  by  frac- 
tions, since  any  whole  number  may  be  divided  into  any  number 
of  parts.  For  example,  6  is  the  same  as  £  .  It  is  also  the  sam,e 
as  J/>  •¥•>  •¥">  s*~i  an(l  an  infinite  number  of  other  expressions 
which  all  have  the  same  value. 

ADDITION   AND   SUBTEACTIOK   OF  FRACTIONS. 

When  fractions  have  the  same  denomination  there  is  no 
more  difficulty  in  adding  or  subtracting  them  than  there  is  in 
adding  or  subtracting  whole  numbers.  Thus  i+-f  is  manifestly 
•f,  and  -f-  —  $  is  obviously  f  .  So  also 

rkr  +  TJhr-^fr-  i 


Also  i  +  f=f=l  and  f  -f  +  J=£=0. 

But  when  fractions  have  not  the  same  denominators,  then, 
before  we  can  add  or  subtract  them,  we  must  change  them  for 
others  of  equal  value  which  have  the  same  denominators.  For 
example,  if  we  wish  to  add  the  fractions  -J  and  -J,  we  must  con- 
sider that  |-  is  the  same  as  f,  and  that  ^  is  equivalent  to  f  .  We 
have,  therefore,  instead  of  the  fractions  first  proposed,  the  equiva- 
lent fractions  f  and  f  ,  the  sum  of  which  is  £.  If  the  two  fractions 
were  united  by  the  sign  —  ,  we  should  have  -J  —  J  or  f  —  f  =  •&• 
Again,  if  the  fractions  proposed  be  •!+-£,  then  as  f  is  the 
same  as  f,  the  sum  will  be  |-  +1=J8L==  -HK  ^  ^e  sum  °f 
•J-  and  J  were  required,  then  as  i=-jV  anc^  i==iV>  ^he  sum 


These  cases  are  simple  and  easily  reduced.  But  we  may 
have  a  great  number  of  fractions  to  reduce  to  a  common  denom- 
ination, which  require  a  more  elaborate  process.  For  example, 
we  may  have  •&,  f  ,  f  ,  |,  -|,  to  reduce  to  a  common  denomination, 
in  order  that  we  may  add  them  together.  The  solution  of  such 
a  case  depends  upon  finding  a  number  that  shall  be  divisible  by 


TO  REDUCE  FRACTIONS  TO  A  COMMON  DENOMINATION.    35 

all  the  denominators  of  these  fractions.    Here  we  proceed  ac- 
cording to  the  following  rule : 

TO   REDUCE  FRACTIONS  TO   A   COMMON  DENOMINATION. 

KFLE. — Multiply  each  numerator  into  every  denominator 
except  its  own  for  a  new  numerator,  and  multiply  all  the  denom- 
inators together  for  a  common  denominator. 

When  this  operation  has  been  performed,  it  will  be  fonnd 
that  the  numerator  and  denominator  of  each  fraction  have  been 
multiplied  by  the  same  quantity,  and  consequently  that  the  frac- 
tions retain  the  same  value,  while  they  are  at  the  same  time 
brought  to  a  common  denomination. 

Example.  Eednce  ^,  f,  f,  4,  and  £,  to  a  common  denomina- 
tion. 

Ix3x4x5x  6=360  and  360—6=  60  and  60—2=30 
2x2x4x5x6=480  and  480— 6=  80  and  80—2=40 
3x2x3x5x6=540  and  540— 6=  90  and  90—2=45 
4x2x3x4x  6=576  and  576—6=  96  and  96—2=48 
5x2x3x4x  5=600  and  600—6=100  and  100—2=50 


2x3x4x5x  6=720  and  720-7-6=120  and  120—2=60 

Here,  then,  we  first  multiply  1,  which  is  the  numerator  of 
the  fraction  £,  by  the  denominators  of  all  the  other  fractions  in 
succession.  We  next  multiply  the  number  2,  which  is  the  nu- 
merator of  the  fraction  f ,  by  the  denominators  of  all  the  other 
fractions — excepting  always  its  own  denominator — and  we  pro- 
ceed in  this  manner  through  all  the  fractions  whatever  their 
number  may  be.  We  next  multiply  all  the  denominators  to- 
gether for  the  common  denominator.  Proceeding  in  this  way 
we  find  the  first  numerator  to  be  360,  the  second  480,  the  third 
640,  the  fourth  576,  and  the  fifth  600 ;  while  the  new  denomi- 
inator  we  find  to  be  720.  It  is  clear,  however,  that  these  frac- 
tions are  not  in  their  lowest  terms,  and  that  the  numerator  and 
denominator  of  each  may  be  divided  by  some  common  number 
without  leaving  a  remainder.  We  may  try  6  as  such  a  divisor, 
and  we  shall  find  that  the  numerators  will  then  become  60,  80, 
90,  96,  and  100,  and  the  denominator  120.  These  numbers, 


36  ARITHMETIC    OF   THE    STEAM-ENGINE. 

however,  are  still  divisible  by  2,  and  performing  the  division 
the  numerators  become  30,  40,  45,  48,  and  50,  and  the  denomi- 
nator becomes  60.  The  same  result  would  have  been  attained 
if  we  had  divided  at  once  by  12.  And  as  we  cannot  effect  any 
further  division  upon  all  of  the  numbers  by  one  common  num- 
ber, without  leaving  a  remainder  in  the  case  of  some  of  them, 
the  fractions,  we  must  conclude,  are  now  in  their  lowest  com- 
mon terms.  To  add  together  these  fractions  we  have  only  to 
add  together  the  numerators,  and  place  the  common  denomina- 
tor under  the  sum.  Performing  this  addition  we  find  that  in 
this  case  we  have  \\^,  and  as  |$  are  equal  to  1,  it  follows  that 
%*•  are  equal  to  3  and  ||,  or  3  f J. 
It  is  easy  to  prove  that  the  fractions 

i,  I,  f ,  *,  and  | 
are  of  precisely  the  same  value  as  the  fractions 

50     4$.     4_5     AS.    $-0 
605    60>    60»    605    60 

which  have  been  substituted  for  them.  Dividing  numerator  and 
denominator  of  the  first  term  by  30  we  obtain  £ ;  dividing  nu- 
merator and  denominator  of  the  second  term  by  20  we  obtain 
f ;  15  is  the  divisor  in  the  case  of  the  third  term  when  we  ob- 
tain f ;  12  is  the  divisor  in  the  case  of  the  fourth  term  when 
•we  obtain  the  fraction  £ ;  and  10  is  the  divisor  in  the  last  case 
when  we  obtain  the  fraction  £ .  Dividing  the  numerator  and 
denominator  of  each  of  the  transformed  fractions,  therefore,  by 
the  greatest  number  that  will  divide  both  without  a  remainder, 
we  get  the  fractions 

i,  I  f,  1 5  and  | 

which,  it  will  be  seen,  are  the  fractions  with  which  we  set  out, 
and  they  are  now  in  their  lowest  terms,  but  are  no  longer  of 
one  common  denomination.  The  lowest  terms  with  a  common 
denominator  are 

30      40     45     48     or,/l    50 
Tffl-J  TT6)  ffin  fffl->  m(i  -STf 

as  determined  above. 

The  subtraction  of  fractions  from  one  another  is  accom- 
plished by  reducing  them  to  a  common  denomination  as  for  ad- 


ADDITION   AND    SUBTRACTION    OF   FRACTIONS.  37 

dition,  and  then  by  subtracting  the  less  numerator  from  the 
greater.  Thus  if  we  have  to  subtract  f-  from  -f-,  we  must  re- 
duce them  to  a  common  denomination  by  the  process  already 
explained,  when  the  first  becomes  |f,  and  the  second  Jf,  so  that 
•f  exceeds  f  in  magnitude  by  ^  So  also  if  we  have  to  subtract 
|  from  f,  the  first  fraction  becomes  by  the  process  of  reduction 
ff ,  and  the  second  f -j>,  so  that  f  taken  from  £  leaves  ?\. 

As  whenever  the  numerator  of  a  fraction  is  a  larger  number 
than  the  denominator,  the  value  of  the  fraction  is  greater  than 
unity,  and  is  equal  to  unity  when  numerator  and  denominator 
is  the  same,  we  have  only  to  divide  the  numerator  by  the  de- 
nominator to  find  the  number  of  integers  which  the  fraction 
contains.  So  in  subtracting  a  fraction  from  a  whole  number, 
we  must  break  one  or  more  integers  up  into  fractions  of  the 
same  denomination  as  that  which  has  to  be  subtracted.  Thus 
if  we  have  to  take  f  £  from  1,  we  must  instead  of  the  1  wrife 
££,  and  ££  taken  therefrom  obviously  leaves  |g.  If  we  have  to 
add  together  such  sums  as  3£  and  2f,  we  see  at  once  that  the 
whole  numbers  when  added  will  be  5,  and  the  equivalent  frac- 
tions under  a  common  denominator  will  be  f  and  $  or  f ,  which 
is  1£,  so  that  the  total  quantity  will  be  6£. 

The  addition  and  subtraction  of  decimal  fractions  are  per- 
formed in  precisely  the  same  way  as  the  addition  and  subtrac- 
tion of  whole  numbers — the  only  precaution  necessary  being  to 
place  the  decimal  point  in  the  proper  place.  Thus  78963-874+ 
83952-2 +  364-003 +  10000-997"  are  added  together  as  follows: 
78963-874  Here,  beginning  as  in  the  addition  of  whole 

83962-2         numbers  with  the  first  column  to  the  right,  we  find 

364-003     that  7  and  3  are  10  and  4  are  14.    "We  set  down 

10000*99*7 

the  4  beneath  the  column  and  carry  1  to  the  next 

173281-074  column.  Adding  up  the  next  column,  we  find  only 
two  significant  figures  in  it,  and  we  say  1  added  to 
9  makes  10,  which  added  to  7"  makes  17.  We  set  down  the  7  and 
carry  the  1  as  before  to  the  next  column,  which  when  added  up 
we  find  to  be  20.  This  means  20  tenths,  and  we  set  down  the 
0  and  carry  the  2  to  the  next  column  just  as  in  simple  addition. 
So  likewise  in  subtraction,  if  we  take  2-25  from  4-75,  the  result 


38  ARITHMETIC   OF  THE   STEAM-ENGINE. 

will  be  2-50;  or  if  we  take  T79  from  3,  the  result  is  T21.     In 
such  a  case  we  write  the  3  thus : 

3-00  Here  we  write  the  3  with  a  decimal  point  after  it, 

1'W  and  we  add  as  many  ciphers  after  the  decimal  point  as 
~7^j  there  are  decimal  figures  to  be  subtracted,  or  we  suppose 
:  those  ciphers  to  be  added.  This  does  not  alter  the  value 
of  the  3,  as  3  with  no  fractions  added  to  it  is  just  3.  Perform- 
ing the  subtraction  we  say  9  from  10  leaves  1,  and  8  taken  from 
10  leaves  2,  and  2  from  3  leaves  1,  just  as  in  simple  subtraction. 

MtTLTIPLICATION  AND  DIVISION   OF  FRACTIONS. 

If  we  wish  to  multiply  a  fraction  any  number  of  times,  it  is 
clear  that  it  is  only  the  numerator  we  must  multiply.  Thus  if 
we  multiply  -J  of  an  inch  by  3,  it  is  obvious  that  we  shall  get  f 
of  an  inch  as  the  product  of  the  multiplication,  or  -J-  repeated 
3  times.  "We  have  already  seen  that  to  multiply  both  terms  of 
a  fraction  by  any  number  does  not  alter  the  value  of  the  frac- 
tion, and  if  we  were  to  multiply  the  numerator  and  denomina- 
tor of  the  fraction  £  by  3  we  should  get  ^,  which  is  just  the 
same  as  -J-.  Thus  also — 

3  times  \  makes  f  or  \\. 

3  times  \  makes  f  or  1. 

3  times  \  makes  f  or  £. 

4  times  -A-  makes  34  or  1A-  or  1£. 

1 6  1  *  1  *  o 

Instead,  however,  of  multiplying  the  numerator,  we  may 
attain  the  same  end  by  dividing  the  denominator,  and  this  is  a 
preferable  practice  when  it  can  be  carried  out,  as  it  shortens  the 
arithmetical  operation.  Thus  J  multiplied  by  2  is  f  or  \.  But 
by  dividing  the  denominator  of  \  by  2,  we  obtain  the  same 
quantity  of  -|-  at  one  operation.  So  also  if  we  have  to  multiply 
f  by  3  we  obtain  -^-,  or  f .  But  if,  instead  of  multiplying  the 
numerator,  we  divide  the  denominator,  we  obtain  the  f  at  one 
operation.  In  the  same  way  ^f  multiplied  by  6  is  equal  ^,  or  3J. 

Where  the  integer  with  which  the  multiplication  is  per- 
formed is  exactly  equal  to  the  denominator  of  the  fraction,  the 
product  will  be  equal  to  the  numerator.  Thus — 


MULTIPLICATION   OF   FRACTIONS   BY   FRACTIONS.         39 

$•  X  2  =  1 
f  X3=2 

f  x4  =  3 

Having  now  shown  how  a  fraction  may  be  multiplied  by  an 
integer,  the  next  step  is  to  show  how  a  fraction  may  be  divided 
by  an  integer ;  and  just  as  a  fraction  may  be  multiplied  by  di- 
viding the  denominator,  so  may  a  fraction  be  divided  by  multi- 
plying the  denominator.     It  is  clear  that  if  we  divide  half  an 
inch  into  two  parts,  each  of  these  parts  will  be  J  of  an  inch, 
and  we  divide  quarter  of  an  inch  into  two  parts,  each  of  those 
parts  will  be  £  of  an  inch,  so  that  J-i-2  ==J  and  i-4-2  =£,  which 
quantities  we  obtain  by  successively  multiplying  the  denomina- 
tors.   "We  may  accomplish  the  same  object  by  dividing  the  nu- 
merator where  it  is  divisible  without  a  remainder.     Thus  f  di- 
vided by  2  is  clearly  £,  and  f  divided  by  3  is  f .     Thus  also 
££  divided  by  2  gives  ^-, 
|f  divided  by  3  gives  ^, 
\\  divided  by  4  gives  ^. 

When  the  numerator  is  not  divisible  by  the  divisor  without 
a  remainder,  the  fraction  may  be  put  into  some  equivalent  form, 
when  the  division  may  be  effected.  Thus  if  we  had  to  divide 
f  by  2,  we  might  turn  it  into  the  equivalent  fraction  f,  which, 
divided  by  2,  gives  f .  But  the  same  number  is  more  conven- 
iently found  by  multiplying  the  denominator  instead  of  by  di- 
viding the  numerator. 

We  have  next  to  consider  the  case  where  one  fraction  has  to 
be  multiplied  by  another.  Thus  if  the  fraction  f  has  to  be  mul- 
tiplied by  the  fraction  f ,  we  have  first  to  remember  that  the  ex- 
pression f  means  2  divided  by  3,  and  we  may  first  multiply  by  4, 
which  produces  £,  and  then  divide  by  5,  which  produces  T^. 
Hence,  in  multiplying  a  fraction  by  a  fraction,  we  multiply  the 
numerators  together  for  the  new  numerator,  and  the  denominat- 
ors together  for  the  new  denominator.  Thus, 

J  x  f  gives  the  product  f  or'£, 

fxi  gives  A> 

J  x  ^  gives  £|  or  V>6, 


40  ARITHMETIC    OF   THE    STEAM-ENGINE. 

Finally,  we  have  to  show  how  one  fraction  may  be  divided 
by  another.  If  the  two  fractions  have  the  same  number  for  a 
denominator,  the  division  takes  place  only  with  respect  to  the 
numerators.  An  inch  being  -^  of  a  foot,  it  is  clear  that  -fy  is 
contained  in  T9^  just  as  often  as  3  inches  is  contained  in  9  inches 
or  3  times ;  and  in  the  same  manner,  in  order  to  divide  -fa  by 
•fs,  we  have  only  to  divide  8  by  9,  which  gives  f .  So  also  /ff  is 
contained  3  times  in  4-®,  and  T^  9  times  in  T4(i9ff.  But  when  the 
fractions  have  not  the  same  denominator,  then  we  must  reduce 
them  to  a  common  denominator  by  the  method  of  reduction  al- 
ready explained.  This  result,  expressed  in  words,  will  be  as 
follows : — Multiply  the  numerator  of  the  dividend  by  the  denom- 
inator of  the  divisor  for  the  new  numerator,  and  the  denomi- 
nator of  the  dividend  by  the  numerator  of  the  divisor  for  the  new 
denominator.  Thus  -f  divided  by  f = y|,  and  £  divided  by  £=f  or 
-f,  or  1J,  and  f-f  divided  by  £ =££!}-  or  -f.  This  rule  is  commonly 
expressed  in  the  following  form : — Invert  the  terms  of  the  divisor 
so  that  the  denominator  may  be  in  the  place  of  the  numerator. 
Multiply  the  fraction  which  is  the  dividend  by  this  inverted  frac- 
tion, and  the  product  will  be  the  quotient  sought. 

Thus  f  divided  by  \— £  x  f  -  £ =1£.  Also,  £  divided  by  £=£  x 
f=H,  and  £|  divided  by  |^|f  x|=r||g  or  f. 

If  we  have  a  line  100  feet  long,  and  if  we  divide  it  in  half,  we 
shall  manifestly  have  two  lines  each  50  feet  long.  So  if  we  di- 
vide it  into  lengths  of  25  feet,  we  shall  have  4  such  lengths ;  if 
we  divide  it  into  lengths  of  2  feet  each,  we  shall  have  50  such 
lengths ;  and  if  into  1  foot  lengths,  we  shall  have  100  of  them ; 
if  into  lengths  of  half  a  foot,  we  shall  have  200  lengths ;  and  if 
into  lengths  of  \  of  a  foot,  we  shall  have  400  such  lengths. 
Hence, 

100  divided  by  100=1 

100  divided  by    60=2 

100  divided  by    25=4 

100  divided  by     1      100 

100  divided  by     £  =  200 

1 00  divided  by     J  =  400 
We  see,  therefore,  that  to  divide  a  number  by  the  fraction  \ 


DIVISION    OF   FRACTIONS    BY   FRACTIONS.  41 

is  equivalent  to  multiplying  it  by  2  ;  to  divide  by  the  fraction  J 
is  the  same  as  to  multiply  by  4.  So,  further,  if  we  divide  1  by 
the  fraction  -j-jjVfr,  the  quotient  is  1,000,  and  1  divided  by  ^^-5-5- 
is  10,000.  As,  then,  the  fraction  gets  smaller  and  smaller,  the 
quotient  gets  greater  and  greater,  so  that  we  are  enabled  to  con- 
ceive that  a  number  divided  by  0  will  be  indefinitely  great,  since 
in  fact  there  will  be  an  indefinitely  great  number  of  nothings  in  it. 
As  every  number  whatever,  divided  by  itself,  produces  unity, 
so  a  fraction,  divided  by  itself,  produces  unity.  Thus  £  -s-f  =f  x 

1  =  1. 

The  multiplication  of  decimal  fractions  is  performed  in  pre- 
cisely the  same  way  as  the  multiplication  of  whole  numbers,  and 
we  must  mark  off  in  the  product  as  many  decimal  places  as  there 
are  in  the  multiplier  and  multiplicand  together.  Thus  1-0025 
multiplied  by  2-5  =  2-50625  ;  also,  '0048  multiplied  by  -000012 
=  0000000576. 

The  division  of  decimals  is  performed  in  the  same  way  as  the 
division  of  common  numbers;  and  if  the  number  of  decimal 
places  in  the  divisor  be  the  same  as  in  the  dividend,  the  quotient 
thus  obtained  will  be  the  quotient  required,  and  will  be  a  whole 
number.  But  if  the  number  of  decimals  in  the  dividend  exceed 
that  in  the  divisor,  mark  off  in  the  quotient  obtained  by  this  di- 
vision as  many  decimal  places  as  make  up  the  difference.  But 
if  the  number  of  decimals  in  the  divisor  exceed  that  in  the  divi- 
dend, annex  as  many  ciphers  to  the  quotient  as  make  up  the  dif- 
ference. Thus  -805  divided  by  2-3  =  '35,  and  2-5  divided  by 
•32  =  7-8125. 

The  number  3'045  denotes  3  units,  0  tenths,  4  hundreths,  and 
5  thousandth,  and  it  might  be  written  3+  -fs+^-y  +Tt&rsi  an(l 
the  number  3  -47  might  be  written  3  +A+r^>or  ^  might  be 


written  300+40  +  7=347      g     ^    m5  =  13^  =  13f   and 
100          100 


23-0625  =  23^^  =  23^.  Also,  4-35  =  4+  A+T*T,  or  to  *  + 
>  or  by  reducing  the  fractions  to  the  same  denomination 
o  +T£ff=£jj-£.  So  flHK  put  in  the  form  of  a  decimal, 
will  be  5-62,  for  |^^^+^+T^.  But  H*=l,  and  there- 
fore  =5 


42  ARITHMETIC   OF   THE    STEAM-ENGINE. 

PROPORTION. 

The  Proportion  or  Ratio  of  one  quantity  to  another  is  the 
number  which  expresses  what  fraction  the  former  is  of  the  lat- 
ter, and  is  therefore  obtained  by  dividing  the  former  by  the 
latter. 

The  most  distinct  idea  of  proportion  is  obtained  by  reference 
to  a  triangle  such  as  that  here  figured,  where  AB  has  the  same 
proportion  to  BO  that  AD  has  to  DE.  It  is  clear  that  if  the  quan- 
tities AB,  AD,  and  BO  are  fixed,  the  quantity  DE  will  also  be  de- 
termined, as  we  have  only  to  draw  the  line  AE  through  o  until 

Fig.  l. 


it  intersects  the  vertical  line  DE,  which  it  will  thereby  cut  off  to 
the  proper  length.  Thus  also  the  ratio  108  to  144,  or  as  it  is 
written  108  :  144,  is  |-£ £  =  f .  A  proportion  is  usually  stated  as 
follows :  2  is  to  4  as  4  is  to  8,  or  2  :  4 : :  4  :  8 ;  and  in  all  cases 
of  proportion  the  product  of  the  first  and  fourth  terms  are  equal 
to  the  product  of  the  second  and  third  terms.  This  is  expressed 
by  saying  that  the  product  of  the  extremes  is  equal  to  the  prod- 
uct of  the  means.  So  2  x  8  =  4  x  4.  Conversely,  if  the  product 
of  any  two  numbers  equal  the  product  of  other  two,  then  the 
four  numbers  are  proportionals.  The  method  by  which  we  find 
a  fourth  proportional  to  three  given  quantities,  by  multiplying 
together  the  second  and  third  and  dividing  by  the  first,  is  what 
is  termed  the  KTJLE  OF  THEEE.  If  a  yard  of  calico  costs  1  shil- 
ling, it  is  clear  that  20  yards  will  cost  20  shillings ;  and  we  say, 
therefore,  1  yard  is  to  20  yards  as  1  shilling  is  to  20  shillings ; 
or  we  say,  3  inches:  12  inches  ::  12  inches:  48  inches.  Here 
we  obtain  the  48  by  multiplying  together  12  and  12,  which 
makes  144,  and  which  divided  by  3  gives  48. 

Proportion  is  in  fact  a  mere  question  of  scale.    If  we  make 


NATURE    OF   PROPORTION. 


43 


a  model  or  drawing  of  a  house  or  a  machine,  we  may  make  it  on 
the  scale  of  J  of  an  inch  to  the  foot,  or  •£  an  inch  to  the  foot,  or 
1  inch  to  the  foot,  or  1£  inches  to  the  foot,  or  on  any  scale  what- 
ever. But  the  object,  when  constructed  of  the  full  size,  will  be 
precisely  the  same  on  whatever  scale  the  model  or  drawing  has 
been  formed.  If  the  scale  be  J  of  an  inch  to  the  foot,  then  it  is 
clear  the  object  when  formed  of  full  size  will  be  48  times  larger 
than  the  model  or  drawing — that  is,  it  will  be  48  times  longer, 
48  times  broader,  and  48  times  higher.  So  in  like  manner  if  the 
•J-  inch  scale  be  employed,  the  object  will  be  24  times  larger;  if 
the  scale  be  1  inch,  it  will  be  12  times  larger ;  and  if  the  scale 
be  1-J-  inches  to  the  foot,  it  will  be  8  times  larger.  So  in  like 
manner  £20  bears  the  same  proportion  to  £1  that  20  shillings 
bears  to  1  shilling.  But  £20  are  400  shillings,  and  £1  are  20 
shillings.  Hence,  by  transforming  the  pounds  into  shillings,  we 
see  that  400  shillings  bear  the  same  relation  to  20  shillings  that 
20  shillings  bear  to  1  shilling;  or,  in  other  words,  400 :  20  :: 
20:  1. 

If  -we  take  a  rectangular  figure  such  as  ABCD,  say  4  inches 
long  and  1  inch  wide,  and  if  we  enlarge  this  figure  by  making  it 
4  inches  longer  and  4  inches  broader,  we  see  at  a  glance  that  the 
resulting  rectangle  AEFG  is  not  of  the  same  shape,  and  in  fact  is 
not  the  same  kind  of  figure  as  the  original  rectangle  ABOD.  This 


is  because  the  enlargement  was  not  made  proportionally,  and 
the  diagonal  AF  consequently  does  not  lie  in  the  same  line  as  the 


44  ARITHMETIC    OF   THE    STEAM-ENGINE. 

diagonal  AO.     To  make  the  enlargement  proportional,  we  should 
only  have  extended  AB  1  inch,  when  we  extended  AD  4  inches. 

Fig.  3. 


Such  an  extension  is  shown  by  the  rectangle  AIHG;  and  the 
diagonal  of  that  rectangle  lies  in  the  same  line  as  that  of  the 
original  rectangle  ABCD.  In  like  manner,  if  the  elliptical  figure 
AB  be  enlarged  by  equal  quantities  in  the  line  AB  and  in  the  line 
CD,  each  successive  ellipse  becomes  more  circular,  and  to  main- 
tain the  original  figure  the  enlargements  should  be  in  the  pro- 
portion of  the  length  and  breadth. 

ON  THE   SQUARES   AND   SQUARE  BOOTS   OP  NUMBERS. 

The  product  of  a  number  multiplied  by  itself  is  called  a  square, 
and  the  quotient  obtained  by  dividing  this  product  by  the  num- 
ber is  the  square  root  of  the  product.  Thus  12  times  12  is  144, 
which  is  the  square  of  12 ;  and  144  divided  by  12  is  12,  which  is 
the  square  root  of  144.  In  like  manner,  the  square  root  of  12  is 
the  particular  number  which,  multiplied  by  itself,  produces  12. 
Such  number  is  neither  3  nor  4,  as  3  tunes  3  is  9  and  4  times  4 
is  16,  of  which  the  one  is  less  than  12  and  the  other  greater. 
The  square  root  of  12  will  be  some  number  between  3  and  4,  and 
what  the  particular  number  is  it  is  the  object  of  the  process  for 
determining  square  roots  to  discover.  The  origin  of  the  term  is 
traceable  to  the  language  of  geometry,  where  a  rectangular  sur- 
face is  produced  by  the  multiplication  of  one  linear  dimension 


SQUARES   OF   INTEGERS   AND   FRACTIONS.  45 

with  another,  or  a  square  is  produced  by  the  multiplication  of 
one  linear  dimension  by  itself.  Thus  a  piece  of  board  a  foot  long 
and  a  foot  broad  has  a  surface  of  one  square  foot,  or,  if  we  count 
the  dimensions  in  inches,  as  the  length  is  12  inches  and  the 
breadth  12  inches,  the  superficies  will  be  12  tunes  12,  or  144 
square  inches.  The  square  of  1  is  1,  since  1  x  1=1.  The  square 
of  2  is  4,  since  2  x  2=4.  The  square  of  3  is  9,  since  3x3=9. 
Contrariwise  1,  2,  and  3  are  the  square  roots  of  1,  4,  and  9. 
If  we  write  the  numbers 

1,  2,  3,  4,  5,  6,  7,  8,  9,  10,  11,  12,  13, 
and  their  squares 

1,  4,  9,  16,  25,  36,  49,  64,  81,  100,  122,  144,  169, 

it  will  be  seen  that  if  each  square  number  is  subtracted  from  that 
which  immediately  follows,  we  obtain  the  series  of  odd  numbers 

3,  5,  7,  9,  11,  13,  15,  17,  19,  21,  &c., 

in  which  the  numbers  go  on  increasing  by  2. 

The  square  of  a  fraction  is  obtained  by  multiplying  the  frac- 
tion by  itself,  in  the  same  manner  as  a  whole  number.  Thus 
ixi=i;  *xi=i;  fxf=$;  ixJ=TV;  and  ixl=-fr.  So 
also  %  is  the  square  root*  of  J;  -J  is  the  square  root  of  £,  and  J  is 
the  square  root  of  T'ff. 

"When  the  square  of  a  mixed  number,  consisting  of  an  integer 
and  a  fraction,  has  to  be  determined,  we  may  reduce  the  mixed 
number  to  a  fraction  by  multiplying  the  integer  by  the  denomi- 
nator, and  adding  the  numerator  to  form  a  new  numerator  with 
the  same  denominator  for  the  denominator  of  the  new  fraction. 
Thus  3i=-y-+£— Y-  and  the  square  of  -3^=-9Al  <>r  15irV  Thtls 
also,  as  the  square  of  -f  is  f|f,  the  square  root  of  f  f  is  |,  and  the 
square  root  of  12£  or  -4/-=|r=3£.  But  when  the  number  is  not 
a  square,  it  is  impossible  to  extract  its  square  root  precisely, 
though  the  root  may  be  approximated  to  with  any  required  de- 
gree of  nearness.  "We  have  already  seen  that  the  square  root  of 
12  must  be  more  than  3  and  less  than  4.  "We  have  also  seen 
that  this  root  is  less  than  3 £,  as  the  square  of  3£  is  12£.  Neither 
is  the  root  3T"V  or  ff  the  square  of  which  is  VaV-  or  12stj>  which 


46  ARITHMETIC    OF   THE    STEAM-ENGINE. 

is  still  greater  than  12.  So  if  we  try  the  number  8/3  or  •^j-'V2/, 
we  shall  find  the  number  to  be  too  small,  for  12  reduced  to  the 
same  -denomination  is  -2yV288'>  so  that  8/3  is  Tg-g  too  small,  while 
3T7j  is  too  great.  The  fact  is,  whatever  fraction  we  annex  to  3, 
the  square  of  that  sum  will  always  contain  a  fraction,  and  will 
never  be  exactly  12;  and  although  we  know  that  3T77  is  too  great, 
and  3Tfi3-  is  too  small,  we  cannot  fix  upon  any  intermediate  num- 
ber which  multiplied  by  itself  shall  produce  12  ;  whence  it  fol- 
lows that  the  square  root  of  12,  though  a  determinate  magnitude, 
cannot  be  expressed  by  fractions.  There  is  therefore  a  kind  of 
numbers  which  cannot  be  specified  by  fractions,  but  which  still 
are  determinate  quantities,  and  of  these  numbers  the  square  root 
of  12  is  an  example.  These  numbers  are  called  irrational  num- 
bers, and  they  occur  whenever  we  attempt  to  find  the  square  root 
of  a  number  that  is  not  a  square.  These  numbers  are  also  called 
surds  or  incommensurables.  The  square  roots  of  all  numbers 
which  are  not  perfect  squares,  are  indicated  by  the  sign  ^/,  which 
is  read  square  root.  Hence  -^/12  means  the  square  root  of  12; 
^/2  the  square  root  of  2 ;  ^/3  the  square  root  of  3  ;  ^/f  the  square 
root  of  •§,  and  «Ja  the  square  root  of  a.  As,  moreover,  the  square 
root  of  a  number  multiplied  by  itself  will  produce  the  number, 
</2  multiplied  by  ^/2  will  produce  2 ;  ^/3  x  ^/3=3  ;  ^/5  x  ^/5=5 ; 
V$  x  V$~$  5  an<^  Vffl  x  Va  produces  a. 

Although  these  irrational  quantities  cannot  be  expressed  in 
fractions,  it  will  not  therefore  be  supposed  that  they  are  visionary 
or  impossible.  On  the  contrary,  they  are  real  quantities,  which 
may  be  dealt  with  in  the  same  way  as  common  numbers ;  and 
however  difficult  of  appreciation  such  a  number  as  the  square 
root  of  12  may  be,  we  at  least  know  this  much  of  it,  that  it  is 
such  a  number  as  multiplied  by  itself  will  produce  12. 

It  is  easy  to  approximate  to  the  square  root  of  a  number  by 
taking  a  trial  number  and  squaring  it,  when  it  will  be  at  once 
seen  whether  such  supposititious  number  is  too  great  or  too  small. 
It  is  also  easy  to  find  the  square  root  by  means  of  logarithms. 
But  the  ordinary  arithmetical  process  for  finding  the  square  root 
is  not  difficult,  and  will  be  readily  understood  by  one  or  two  ex- 
amples. 


SQUARE   ROOTS   OF   NUMBERS.  47 

Thus,  in  extracting  the  square  roots  of  15,625  and  998,001, 
the  mode  of  procedure  is  as  follows : — 


15625(125  •  998001(999 

1  81 

22)56  189)1880 

44  1701 


245)1225  1989)17901 

1225  17901 


Here,  in  the  first  place,  we  separate  the  numbers  into  groups 
of  two  figures  each,  beginning  at  the  right,  by  making  a  short 
line  over  each  pair  of  figures,  or  by  pointing  them  off  into  groups 
by  such  point  or  mark  as  shall  not  be  confounded  with  the  deci- 
mal point.  "We  then  find  the  next  lowest  square  of  the  first 
group,  which  we  set  under  that  group,  and  subtract  as  in  long 
division,  setting  the  quotient  in  the  usual  place  according  to  the 
mode  of  procedure  in  that  process.  "We  next  double  the  quotient 
for  the  next  trial  divisor,  and  the  quotient  which  we  think  we 
shall  obtain  we  also  place  in  the  divisor,  of  which  it  forms  a  con- 
stituent part;  and  dividing  by  the  divisor  thus  increased,  we 
perform  the  division,  setting  the  quotient  in  the  usual  place  as  in 
long  division.  We  then  subtract,  and  for  the  next  trial  divisor 
we  use  the  first  term  of  the  last  divisor,  and  double  the  last  term 
of  the  quotient.  In  the  first  example,  consisting  of  five  figures, 
we  have  only  one  figure  in  the  first  group,  and  that  figure  is  1. 
Now  the  square  root  of  1  is  1,  which  number  we  set  in  the  quo- 
tient, and  double  it  for  the  next  trial  divisor,  which  therefore 
becomes  2 ;  and  as  2  will  go  twice  in  5,  we  set  2  in  the  quotient, 
and  also  add  it  to  the  trial  divisor  to  make  the  true  divisor ; 
and  so  on.  In  the  second  example,  the  first  group  consists  of 
the  figures  99,  the  nearest  square  to  which  is  81,  and  we  there- 
fore set  9  in  the  quotient,  and  put  twice  9,  or  18,  for  the  next 
trial  divisor,  and  we  see  that  the  number  to  be  added  thereto  to 
exhaust  the  dividend  must  be  large,  as  18  is  contained  10  times 
in  188.  The  number  to  be  added  to  the  trial  divisor  we  find  to 


48  AKITHMETIC    OF   THE    STEAM-ENGINE. 

be  9,  and  we  set  it  in  the  quotient,  and  double  it  to  add  to  the 
first  trial  divisor  to  form  the  second  trial  divisor;  and  so  on 
through  all  the  terms,  bringing  down  at  each  stage  a  group  of 
two  figures,  instead  of  a  single  figure,  as  in  long  division.  When 
there  is  a  remainder  after  all  the  figures  have  been  brought  down, 
the  number  is  not  a  complete  square,  and  its  exact  root  cannot 
be  found,  but  it  may  be  approximated  to  by  using  decimals  to 
carry  on  the  division  with  sufficient  nearness  for  all  useful  pur- 
poses. 

ON  THE  CUBES  AND  CUBE  BOOTS  OF  NUMBEES. 

When  any  number  is  multiplied  twice  by  itself,  or,  what  is 
the  same  thing,  when  the  square  of  a  number  is  multiplied  by 
the  number,  the  product  is  the  cube  of  the  number.  Thus 
2x2x2=8,  and  8  therefore  is  the  cube  of  2.  Also  4  is  the  square 
of  2,  and  4  x  2=8.  In  like  manner,  3  x  3  x  3=27,  and  27  is  the 
cube  of  3  ;  4  x  4  x  4=64,  and  64  is  the  cube  of  4 ;  a  x  «  x  a=a\ 
and  a?  is  the  cube  of  a  ;  or  <z2  x  a=a?.  The  cubes  of  the  first 
nine  numerals  are  1,  8,  27,  64,  125,  216,  343,  512,  and  729,  and 
the  respective  differences  of  these  numbers  are  7,  19,  37,  61, 
127,  169,  217,  271,  where  we  do  not  discern  any  law  of  increase. 
But  if  we  take  the  respective  differences  of  these  last  numbers, 
we  obtain  the  numbers  12,  18,  24,  30,  36,  42,  48,  54,  60,  where 
it  is  evident  that  the  addition  of  the  number  6  to  each  successive 
term  produces  the  next  one. 

In  the  cubes  of  fractions  the  same  law  holds  as  in  the  case 
of  the  squares  of  fractions.  Thus  as  the  square  of  -J-  is  J,  so  the 
cube  of  i  is  £.  So  also  -fa  is  the  cube  of  -J ;  -^  is  the  cube  of  f , 
and  1 1  is  the  cube  of  £ . 

In  the  case  of  the  cubes  of  mixed  numbers,  we  first  reduce 
those  mixed  numbers  to  an  improper  fraction,  and  then  cube 
them  as  above.  Thus  the  cube  of  1-jr  is  the  same  as  the  cube  of 
|,  which  is  -V-  or  3f,  and  the  cube  of  3J  or  J£  is  2Jf-7,  or  34f£. 

The  cube  of  a  &  is  a:i  &3,  whence  we  see  that  if  a  number  has 
factors,  we  may  find  its  cube  by  multiplying  together  the  cubes 
of  the  factors.  Thus  the  cube  of  12  is  1728.  But  12  is  com- 
posed of  the  factors  3  and  4;  and  the  cube  of  3  is  27,  and  the 


CUBES  AND  CUBE  ROOTS  OF  NUMBERS.        49 

cube  of  4  is  64.  Hence  27  x  64=1728  will  be  the  cube  of  12,  as 
by  multiplying  12  by  itself  twice  it  is  found  to  be.  The  cube  of  a 
positive  number  will  always  be  positive,  and  of  a  negative  num- 
ber, negative.  This  is  obvious  on  considering  that  +ax  +ax 
+  a=+a?,  and  that  —  ax—  a~  +  a?,  and  this  multiplied  again 
by  —a  produces  —  a?.  So  the  cube  of  — 1  is  —1,  the  cube  of  —2 
is  — 8,  the  cube  of  — 3  is  — 27,  and  so  of  all  negative  numbers. 

The  cube  root  of  a  number  is  expressed  by  the  sign  ^/,  and  it 
is  easy  to  determine  the  cube  root  of  a  number  when  the  num- 
ber is  really  a  cube.  Thus  we  see  at  once  that  the  cube  root  of 
1  is  1,  that  the  cube  root  of  8  is  2,  that  the  cube  root  of  27  is  3, 
that  the  cube  root  of  64  is  4,  and  that  the  cube  root  of  125  is  5. 
"We  further  see  that  the  cube  root  of  -/T  will  be  f ,  of  fj  will  be 
f,  and  of  2J£,  or  ff ,  is  f  or  \\.  But  if  the  proposed  number  be 
not  a  cube,  it  cannot  any  more  than  in  the  case  of  the  square 
root  be  expressed  accurately,  either  by  whole  or  fractional  num- 
bers, though  an  approximate  expression  may  be  obtained  that 
will  be  sufficiently  near  the  truth  for  all  useful  purposes.  For 
instance,  43  is  not  a  perfect  cube,  and  it  is  impossible  to  specify 
any  number,  whether  whole  or  fractional,  which,  multiplied  by 
itself  twice,  will  produce  43.  If  we  take  a  number  as  nearly  as 
we  can  to  that  which  we  suppose  the  cube  root  should  be,  and 
multiply  it  twice  by  itself,  we  shall  at  once  see  whether  such 
trial  number  is  too  great  or  too  small.  Thus  if  we  fix  upon  8$ 
or  f  as  the  trial  number,  then  we  find  that  the  cube  of  £•  being 
-^f^,  or  42|,  the  number  will  err  in  defect,  42£  being  \  less  than 
43.  By  taking  other  numbers,  we  may  approximate  still  more 
nearly  to  the  true  root,  but  we  shall  never  be  able  to  express  it 
in  figures  precisely,  and — as  in  the  similar  case  in  the  doctrine 
of  square  roots — such  quantities  are  termed  irrational  quantities. 

ON   POWEE8   AND   EOOTS    IN   GENERAL. 

The  product  arising  from  multiplying  a  number  once  or  many 
times  by  itself  is  termed  a  power.  The  square  of  a  number  is 
sometimes  called  its  second  power ;  the  cube  is  sometimes  called 
its  third  power,  and  we  may  have  its  fourth  power,  its  fifth 
pOAver,  or  any  power  depending  on  the  number  of  the  multipli- 
3 


50 


ARITHMETIC    OF   THE    STEAM-ENGINE. 


cations,  or  we  may  say  that  the  number  has  been  raised  to  the 
second,  third,  fourth,  or  fifth  degree.  The  fourth  power  of  a 
number  is  sometimes  called  its  Mquadrate,  but  after  this  degree 
powers  cease  to  have  any  other  than  numerical  appellations. 

It  is  difficult  to  make  the  reason  or  processes  of  the  ordinary 
arithmetical  rule  for  the  extraction  of  the  cube  root  very  intelli- 
gible without  the  aid  of  Algebra,  of  the  processes  of  which  the 
rule  is  only  a  translation.  But  an  example  will  show  the  mode 
of  procedure. 

Let  us  suppose  that  we  had  to  extract  the  cube  root  of  the 
number  80,677,568,161. 


123 


4800 
369 

5169 


1292       554:700 
2584 


557284 


12961  55987200 
12961 


56000161 


80677568161(4321 
64 


16677 


15507 


1170568 


1114568 


56000161 


56000161 


Here  we  first  divide  the  number,  beginning  at  the  right  hand, 
into  groups  of  three  figures  in  each — -just  as  in  extracting  the 
square  root  we  divide  the  number  into  groups  of  two  figures  in 
each.  In  the  last  of  the  groups  we  thus  form  there  happens,  in 
this  example,  to  be  only  two  figures,  and  sometimes  there  will 
be  only  one. 

We  now  consider  what  is  the  next  lower  cube  to  the  number 
80,  and  we  find  that  it  is  64,  which  is  the  cube  of  4.  We  set 
the  figure  4  in  the  quotient,  and  subtract  its  cube  64  from  80, 
which  leaves  a  remainder  of  16.  We  next  bring  down  the  fol- 
lowing period  677. 


MODE    OF   FINDING   THE    CUBE   KOOT.  51 

The  next  step  is  to  set  the  triple  of  the  first  figure  of  the  root 
(12)  at  some  distance  to  the  left  of  the  remainder.  (There  is  123 
in  the  sum,  but  the  3  will  be  accounted  for  presently.)  We  then 
multiply  this  triple  by  the  first  figure  of  the  root,  and  place  the 
product  48  between  the  12  and  the  remainder,  annexing  two 
ciphers  to  it. 

"We  now  divide  the  remainder  by  this  4800,  as  a  trial  divisor, 
and  set  the  quotient  3  as  the  second  figure  in  the  root,  and  also 
after  the  12,  making  123.  "We  next  multiply  this  123  by  3,  the 
second  figure  of  the  root,  set  the  product  369  under  the  4800, 
and  add  them  together.  The  resulting  sum,  5169,  is  the  first 
real  divisor.  "We  next  multiply  the  divisor  by  the  second  figure 
of  the  root,  and  subtract  the  product  15507,  as  in  long  division, 
bringing  down  the  next  period  568. 

To  obtain  the  next  real  divisor  we  proceed  as  follows : — We 
first  triple  the  last  figure  3,  of  123,  which  gives  129.  (There  is 
1292  put  down,  but  the  last  figure,  2,  will  be  accounted  for  pres- 
ently.) The  other  quantity,  5547,  is  found  by  adding  9,  the 
square  of  the  second  figure  of  the  root,  to  the  two  preceding 
middle  lines,  369  and  5169.  We  now  add  two  ciphers  and  re- 
peat the  whole  process,  and  we  find  the  next  figure  of  the  root 
to  be  2,  which  is  the  2  added  to  the  129. 

In  the  case  of  decimals  occurring  in  any  number  of  which  we 
have  to  extract  the  cube  root,  the  distribution  of  the  figures  into 
groups  of  three  each  will  begin  at  the  decimal  point,  and  will 
proceed  to  the  left  for  integers,  and  to  the  right  for  fractions — 
adding  ciphers  where  necessary  to  make  up  the  required  number 
of  figures.  Thus  if  we  had  to  extract  the  cube  root  of  *01,  we 
might  write  the  number  -010,  and  in  like  manner  24'1  might  be 
written  24-100 

It  will  now  be  shown  that  to  add  the  exponents  of  numbers 
is  equivalent  to  multiplying  the  numbers. 

ON  HOOTS  AS  BEPEESENTED  BY  FRACTIONAL  EXPONENTS. 

The  multiplication  or  division  of  numbers  is  indicated  by 
adding  or  subtracting  their  exponents,  and  as  2  may  be  written 


52  ARITHMETIC    OF   THE    STEAM-ENGINE. 

as  21,  then  2*  x  22=2',  since  •&+£  =  !.  As,  too,  the  third, 
fourth,  fifth,  &c.,  powers  of  a  number  are  represented  by  the 
expressions  23,  24,  25,  &c.,  so  the  third,  fourth,  fifth,  &c.,  roots 
are  represented  by  the  expressions  ^2,  ^/2,  ^/2,  &c.  The  square 
root  may  be  written  ty,  or  more  simply  -J.  Now  as  we  have 

seen  that  2*  x  2^  =  2,  and  as  ^/2  x  V2  also  =  2,  it  follows  that 

i  j.  JL 

22  is  another  form  of  expression  for  ^/2.     So  also  2:J=^2,  24  = 

4/2,  25  =  ^2;  and  so  of  all  other  roots  whatever.     Since  also 

21  x  2*  =  2'  =2%  it  follows  that  2  Ms  the  same  as  V22.    In  like 

2.  a. 

manner,  2a=^/23  and  24=^/23. 

When  the  fraction  which  represents  the  exponent  exceeds 
unity,  it  may  either  be  expressed  in  the  form  of  an  improper 
fraction,  or  in  that  of  a  mixed  number.  For  example,  the  frac- 
tion 27  may  be  expressed  in  the  form  2  2.  But  2  2  is  the  prod- 
uct of  22  by  2*,  and  it  may  be  written  in  the  form  2  ^25. 

ON  THE  CLASS  OF  FBACTIONAL  EXPONENTS  TEEMED  LOGABITHMS. 

Since  the  square  root  of  a  given  number  is  a  number  whose 
square  is  equal  to  that  given  number,  and  since  the  cube  root  of 
a  given  number  is  a  number  whose  cube  is  equal  to  that  given 
number,  and  so  of  all  roots  whatever,  it  follows  that  any  number 
whatever  being  given,  we  may  always  suppose  such  roots  of  it 
that,  raised  to  their  respective  powers,  they  shall  always  be  equal 
to  the  given  number.  Since,  also,  powers  with  negative  expo- 
nents are  fractions,  and  powers  with  positive  exponents  are 
whole  numbers,  and  as  all  numbers  whatever  may  be  expressed 
by  whole  numbers  and  fractions,  it  is  clear  that  if  we  take  any 
given  number,  such  as  10,  we  may  raise  it  to  such  a  power  either 
positive  or  negative  as  will  make  it  equal  to  any  number  what- 
ever that  we  may  think  proper  to  assign.  Thus  if  we  fix  upon 
the  number  4,  it  is  certain  that  there  is  a  certain  power  of  the 
number  10,  which  is  equal  to  4.  Or  if  we  fix  upon  the  number 
40,  or  47,  or  57,  or  381,  or  any  other  number  whatever,  then 


NATURE    OF   LOGARITHMS.  53 

there  will  be  some  power  or  other  of  10  that  will  be  equal  to 
those  several  numbers.  Putting  5  for  this  unknown  exponent, 
then  10*=  381,  or  any  other  number  depending  on  the  value  of 
5.  If  instead  of  10  we  write  the  letter  <z,  and  instead  of  381,  or 
a  raised  to  the  power  J,  we  write  the  letter  c,  then  we  obtain 
the  expression  a  =  c.  Here  c  is  the  given  number,  a  is  the  root 
or  radix,  and  5  is  the  exponent  or  logarithm  of  the  number  c 
with  the  radix  a.  The  radix  of  the  common  system  of  logarithms 
is  the  number  10,  and  the  logarithm  of  a  given  number  is  the 
power  to  which  10  must  be  raised  to  be  equal  to  that  given  num- 
ber. Every  number  whatever  has  its  corresponding  logarithm ; 
and  when  we  know  its  logarithm,  we  may,  instead  of  the  num- 
ber, use  the  logarithm,  with  this  conspicuous  advantage,  that 
when  we  have  to  multiply  two  numbers  together  we  shall  ac- 
complish that  end  by  adding  their  logarithms  to  obtain  a  new 
logarithm,  the  number  corresponding  to  which  will  be  the  cor- 
rect product  of  the  two  numbers ;  or  if  we  have  to  divide  one 
number  by  another,  we  shall  accomplish  the  object  by  subtract- 
ing the  logarithm  of  the  one  from  that  of  the  other — the  differ- 
ence constituting  a  new  logarithm,  which  will  be  the  logarithm 
of  the  quotient.  This  quality  of  logarithms  is  apparent  when  we 
recollect  that  they  are  all  exponents  of  a  given  number  a,  and 
that  a*  x  o3=  a5,  or  that  a5  x  <z8=  a13,  where  the  multiplication  is 
signified  by  adding  the  exponents.  So  also  as  a2X3=a6,  #3X3= 
a?,  aSX4=au2,  ffl4XS=a'20,  it  follows  that  to  multiply  a  logarithm 
by  3,  4,  6,  or  any  other  number,  is  equivalent  to  raising  the 
number  to  the  third,  fourth,  fifth,  or  other  corresponding  power ; 
and  contrariwise,  to  divide  the  logarithm  by  3,  4,  5,  or  any  other 
number,  is  equivalent  to  the  extraction  of  the  third,  fourth,  fifth, 
or  any  other  root  of  the  number.  From  these  considerations  it 
will  be  at  once  apparent  that  by  the  use  of  logarithms  an  enor- 
mous amount  of  labour  may  be  saved  in  performing  arithmetical 
computations,  and  to  facilitate  such  computations  the  logarithms 
of  all  the  numbers  usually  occurring  in  calculations  have  been 
ascertained  and  arranged  in  tables,  so  as  to  facilitate  their  em- 
ployment. All  positive  numbers,  such  as  1,  2,  3,  4,  5,  &c.,  are 
logarithms  of  the  root  or  radix  a,  and  of  its  positive  powers,  and 


54  ARITHMETIC    OF   THE    STEAM-ENGINE. 

are  consequently  logarithms  of  numbers  greater  than  unity.  On 
the  contrary,  the  negative  numbers  — 1,  — 2,  — 3,  — 4,  — 5,  &c., 

are  the  logarithms  of  the  fractions  — ,  — ,  — ,  — ,  &c.,  which  are 

a   a-    a:!    a1 

less  than  unity  and  greater  than  nothing.  Now  as  every  signifi- 
cant number  can  only  be  positive  or  negative,  and  as  the  loga- 
rithms of  numbers  greater  than  unity  are  positive,  and  the 
logarithms  of  numbers  less  than  unity  but  greater  than  nothing 
are  negative,  there  is  no  sign  left  to  express  numbers  less  than 
nothing,  or  negative  numbers,  and  we  must  therefore  conclude 
that  the  logarithms  of  negative  numbers  are  impossible. 

It  has  already  been  stated  that  in  the  logarithmic  tables  at 
present  in  common  use,  the  radix,  of  which  the  logarithmic  num- 
ber is  the  exponent,  is  10.  If  we  denote  this  radix  by  a,  then 
the  logarithm  of  any  number  c  is  the  exponent  to  which  we 
must  raise  the  radix  a  or  10,  in  order  that  the  power  result- 
ing from  it  may  be  equal  to  the  number  c.  If  we  denote  the 
logarithm  of  c  by  log.  c,  then  10l°e-e=c.  Now  as  a°—~\.  and 
«'=«,  so  10°=1  and  10' =10.  But  as  the  exponents  are  the 
logarithms  of  the  numbers,  it  follows  that  the  logarithm  of  1  is 
0,  and  the  logarithm  of  10  is  1.  So  also  log.  100  or  10^=2 ;  log. 
1000  or  103=3 ;  log.  10000  or  104=4;  log.  100000  or  1&=5,  and 
log.  1000000  or  106=6.  In  like  manner  log.  TV—  — 1 ;  log. 
T5 o  =  —2 ;  log.  TT\nr=  —3  ;  log.  TJT-J  „  „  =  —4 ;  log.  •n^svv=  — 5  ? 
l°g-  TTnrTo(ro=  — 6?  an<i  so  on  indefinitely. 

Since  log.  1=0  and  log.  10=1,  it  is  plain  that  the  logarithms 
of  all  numbers  between  1  and  10  must  be  less  than  unity  and 
greater  than  nothing.  Let  us  suppose  that  it  was  required  to 
determine  the  logarithm  of  the  number  2.  If  we  represent  this 
logarithm  by  the  letter  a;,  then  we  shall  have  this  expression 
10*=2.  In  order  to  determine  the  value  of  #,  we  may  make  a 
few  approximate  suppositions.  If  we  suppose  x  to  be  -J-,  we  shall 
have  10—2,  which  is  manifestly  too  great,  since  9  _3  and  10* 
must  therefore  be  more  than  3.  If  we  suppose  x  to  be  £,  the 
quantity  will  still  be  too  great.  For  if  10^=2,  then  10^=23,  or 
10'  or  10=8,  which  shows  that  £  is  too  much.  If  we  take  J  as 


MODE   OF   COMPUTING   LOGARITHMS.  55 

1  4 

the  exponent,  then  we  have  10T=2,  or  104=24,  or  10=16,  which 
shows  that  J  is  too  small,  while  £  is  too  great. 

By  pursuing  the  investigation  in  this  manner,  we  should  find 
with  any  required  degree  of  accuracy  what  the  exponent  would 
be  that,  if  10  were  raised  to  that  power,  would  be  equal  to  2. 
This  exponent  or  logarithm,  as  it  is  termed,  would  in  point  of 
fact  be  0-3010300,  or  a  little  less  than  £,  and  in  the  logarithmic 
tables  in  common  use  the  logarithms  are  always  expressed  in 
decimal  fractions,  as  being  the  most  convenient  form  for  pur- 
poses of  computation.  The  value  of  this  decimal  expressed  in 
vulgar  fractions  is  T3n+T§ff  +  T«1o7+T4^+Tiro3ffffff+TTnrro^  + 
TSTS AffoTT-  Logarithmic  tables  are  commonly  computed  to  seven 
places  of  decimals,  as  decimals  carried  to  7  places,  though  not 
expressing  the  result  with  absolute  exactness,  will,  it  is  con- 
sidered, give  results  that  are  sufficiently  accurate  for  all  ordinary 
purposes.  According  to  this  method  of  expressing  logarithms, 
the  logarithm  of  1  will  be  O'OOOOOOO,  since  it  is  really=0.  The 
logarithm  of  10  will  be  I'OOOOOOO,  since  it  is=l.  The  logarithm 
of  100  will  be  written  2-0000000, =2,  and  so  on.  The  logarithms 
of  all  numbers  intervening  between  10  and  100,  and  conse- 
quently composed  of  2  figures,  will  be  greater  than  1  and  less 
than  2,  and  are  expressed  by  1  +  a  decimal  fraction.  Thus  log. 
50=1-6989700.  The  logarithms  of  numbers  between  100  and 
1000  are  expressed  by  2+  a  decimal  fraction;  the  logarithms 
of  numbers  between  1000  and  10,000  are  expressed  by  3+  a 
decimal  fraction.  The  logarithms  of  numbers  between  10,000 
and  100,000  are  expressed  by  4  and  a  decimal  fraction,  and  the 
number  prefixed  to  the  decimal  will  always  be  1  less  than  the 
number  of  figures  in  the  given  number.  Thus  the  logarithm  of 
2290  is  3-3598355,  for  as  there  are  four  figures  in  2290,  the  num- 
ber prefixed  to  the  decimal  will  be  3.  The  number  prefixed  to 
the  decimal,  or  the  integral  part  of  the  logarithm,  is  termed  the 
characteristic  ;  and  when  a  number  consists  of  four  figures,  such 
as  the  number  2290,  its  characteristic  is  invariably  3.  If  the 
number  be  reduced  to  229,  its  characteristic  will  be  2 ;  if  reduced 
to  22  its  characteristic  will  be  1,  and  if  reduced  to  2  its  charac- 
teristic will  be  0.  There  are  therefore  two  parts  to  be  con- 


56  ARITHMETIC    OF   THE    STEAM-ENGINE. 

sidered  in  a  logarithm :  first  the  characteristic,  which  we  can  at 
once  determine  when  we  know  the  number  of  figures  of  which 
the  given  number  consists ;  and  second  the  decimal  fraction, 
which  is  determined  by  the  nature  of  those  figures.  So  also  we 
know,  at  the  first  sight  of  the  characteristic  of  a  logarithm,  what 
is  the  number  of  figures  composing  the  number  of  which  it  is 
the  logarithm.  If  for  example  the  logarithm  6'4771213  be  pre- 
sented, we  know  at  once  that  the  number  of  which  it  is  the 
logarithm  must  consist  of  7  figures,  and  must  be  over  1,000,000. 
The  integral  part  of  a  logarithm  therefore  being  so  easily  found, 
the  main  part  requiring  consideration  is  the  decimal  part,  and  it 
is  that  part  alone  which  is  given  in  the  logarithmic  tables  in 
common  use.  To  show  the  manner  of  using  these  tables,  we 
may  multiply  together  the  numbers  343  and  2401  by  the  aid  of 
logarithms.  Here — 

Log.     343=2-5352941  )     , ,   , 
Log.  2401  =  3-3803922  f  ac 

5-9156863  their  sum. 
Log.  823540=5-9156847  nearest  tabular  log. 

16  difference. 

"We  look  in  the  table  of  logarithms  opposite  the  figures  343, 
and  we  find  the  number  5352941,  which  we  know  constitutes 
the  fractional  part  of  the  logarithm,  while  the  integral  part  will 
be  1  less  than  the  number  of  figures  in  343,  or  in  other  words 
the  integral  part  will  be  2.  In  like  manner  we  find  the  logarithm 
of  2401,  and  adding  these  logarithms  together,  we  find  their 
sum  to  be  5'9156863.  "We  then  look  in  the  table  to  find  the  next 
less  logarithm  to  this,  which  we  find  to  be  5'9156847.  We  see 
at  once  by  the  magnitude  of  the  characteristic  that  the  number 
of  which  this  is  the  logarithm  must  consist  of  six  figures,  and 
we  find  the  number  answering  to  this  logarithm  to  be  823540. 
The  difference  between  the  logarithm  formed  by  the  addition  of 
the  two  original  logarithms  and  its  next  lower  tabular  logarithm 
is  16,  and  in  the  tables  there  is  a  column  of  differences  intended 
to  fix  the  numerical  value  of  such  differences,  and  which  in  this 


COMPUTATION    OF    COMPOUND    QUANTITIES.  57 

case  would  amount  to  the  number  3.     With  this  correction  the 
product  of  343  and  2401  will  become  823543. 

It  is  in  the  extraction  of  roots,  however,  that  logarithms  be- 
come of  the  most  eminent  service.  If,  for  instance,  we  had  to 
extract  the  square  root  of  10,  we  should  only  have  to  divide 
the  logarithm  of  10  which  is  1*0000000  by  2,  which  gives 
O'SOOOOOO  as  the  logarithm  of  the  root  required ;  and  by  refer- 
ring to  the  table  of  logarithms,  we  should  find  that  the  number 
answering  to  this  logarithm  was  3*16228,  which  consequently  is 
the  square  root  of  10.  So  also  if  we  had  to  extract  the  fifth 
root  of  2  we  should  divide  the  logarithm  of  2,  which  is  0'3010300, 
by  5,  which  gives  a  quotient  of  0'0602060,  the  number  answer- 
ing to  which  in  the  tables  is  1'1497,  which  consequently  is  the 
fifth  root  of  2. 

OK   THE   COMPUTATION   OF   COMPOUND   QUANTITIES. 

Hitherto  our  investigations  have  been  restricted  to  the  modes 
of  calculation  suited  to  the  measurement  of  simple  quantities ; 
but  many  of  the  quantities  with  which  we  have  to  deal  in  engi- 
neering practice  are  compound  quantities  made  up  of  simple 
quantities  in  different  forms  of  combination,  and  it  is  now  neces- 
sary to  consider  the  mode  of  computing  the  values  of  these 
compound  quantities.  One  of  the  most  familiar  forms  of  a 
compound  quantity  is  a  sum  of  money  expressed  in  pounds, 
shillings,  and  pence,  or  in  other  coins  of  different  values.  An- 
other variety  is  a  given  weight  expressed  in  tons,  hundred- 
weights, quarters,  and  pounds,  or  in  other  different  kinds  of 
weights.  If  we  wish  to  know  what  number  of  pence  there  is  in 
a  sum  of  money,  or  what  number  of  pounds  or  ounces  there  is  in 
a  given  weight,  the  operation  is  termed  reduction,  and  is  per- 
formed by  multiplying  the  given  quantity  by  the  number  which 
shows  how  many  of  the  next  lower  denomination  makes  one  of 
the  higher.  Thus  if  we  wish  to  know  how  many  pence  there 
are  in  37/.,  we  first  multiply  the  37£.  by  20,  which  will  show  tho 
number  of  shillings  there  are  in  87Z.,  for  as  there  are  20  shillings 
in  II.,  there  will  be  20  times  37  in  37Z.  Now  37x20=740 
3* 


58  ARITHMETIC    OF   THE    STEAM-ENGINE. 

shillings,  and  as  there  are  12  pence  in  1  shilling,  there  will  be 
12  times  740=8880  pence  in  37Z.  If  the  sum  were  37Z.  16s.  8d. 
in  which  we  wished  to  find  the  number  of  pence,  it  is  clear  that 
the  number  of  pence  in  16s.  8d.  must  be  added  to  the  number 
already  found.  Now  as  there  are  12  pence  in  1  shilling,  12 
times  16=192,  the  number  of  pence  in  16  shillings,  to  which,  if 
we  add  the  8  pence  remaining,  we  shall  have  200  pence  to  add 
to  the  8880,  or  in  other  words  we  shall  have  9080  pence  as  the 
answer.  So  if  we  wish  to  ascertain  the  number  of  pounds 
weight  in  3  tons,  we  have  first  to  ascertain  by  a  reference  to  a 
table  of  weights  how  many  pounds  there  are  in  the  ton,  and 
which  we  shall  find  to  be  2240.  This  number  multiplied  by  3 
will  obviously  be  the  number  of  pounds  weight  contained  in  3 
tons.  But  if  the  weight  which  we  were  required  to  find  the 
number  of  pounds  in  were  3  tons  7cwt.  2  quarters  and  8  pounds, 
we  should  first  have  to  multiply  the  3  tons  by  20  to  reduce  them 
to  cwts.,  and  as  there  are  20  cwt.  in  the  ton,  3  tons  would  be 
60  cwt.  But  besides  these  we  have  7  cwt.  more,  so  that  we 
have  in  all  67  cwt.  Now  as  there  are  4  quarters  in  the  cwt., 
there  will  in  67  cwt.  be  4  times  67=268  quarters,  to  which  we 
have  to  add  the  two  quarters  of  the  original  sum,  making  in  all 
270  quarters  in  the  weight.  But  as  there  are  28  Ibs.  in  1  quarter, 
there  will  be  28  times  270=7560  Ibs. ;  and  as  there  are  8  pounds 
besides  to  be  added,  the  sum  total  of  the  weight  will  be  7568  Ibs. 
So  if  we  wished  to  know  how  many  square  inches  there  were  in 
2J  square  feet,  it  is  plain  that  as  there  are  144  square  inches  in  the 
square  foot,  there  will  be  288  square  inches  in  2  square  feet,  and 
36  square  inches  in  J  of  a  square  foot,  and  288  +  36=324  square 
inches.  In  performing  these  and  similar  operations  it  is  of  course 
necessary  to  have  access  to  proper  tables  of  weights  and  measures, 
or,  in  other  words,  to  certain  standard  magnitudes,  as  it  is  impossi- 
ble to  form  an  idea  of  any  magnitude  except  by  comparing  it 
with  some  other  magnitude,  such  as  a  pound,  a  foot,  or  a  gallon, 
of  which  we  have  a  definite  conception. 

On  the  addition  of  compound  quantities. — The  first  step  in 
performing  this  addition  is  to  set  the  quantities  to  be  added  un- 
der one  another,  so  that  terms  of  the  same  kind  may  be  in  the 


COMPUTATION   OF   COMPOUND    QUANTITIES.  59 

same  column.  When  the  relation  between  the  different  quanti- 
ties is  known — as  it  is  in  all  cases  of  arithmetical  addition — we 
add  up  the  numbers  in  the  right-hand  column,  and  divide  by  the 
number  in  this  column  which  makes  1  in  the  next  column.  We 
then  set  the  remainder,  if  any,  under  the  first  column,  and  carry 
the  quotient  to  be  added  to  the  next,  and  so  on  through  all  the 
columns.  Thus  in  adding  up  the  pounds,  shillings,  and  pence 
here  set  down  we  proceed  as  follows : 

We  first  arrange  the  pounds,  shillings,  and  pence  in  three 

»  ,    columns,  with  the  units  under  the  units,  the  tens  un- 

13    0    8    der  the  tens,  and  so  on,  as  in  simple  addition.     We 

256  then  add  up  the  column  of  pence,  and  find  how  many 
37  8  10  Pence  **  contains.  But  as  every  group  of  12  pence 
1297  makes  1  shilling,  we  divide  the  total  number  of  pence 

0  13  4  t>y  12  to  find  how  many  of  such  groups  there  are,  or, 
71  o g  in  other  words,  how  many  shillings  there  are  in  the 
1  total  number  of  pence.  These  shillings  we  transfer 
to  the  shillings  column,  and  as  after  we  have  done  this  there  are 
6  pence  left,  we  write  the  6  beneath  the  pence  column,  and 
then  proceed  to  add  up  the  shillings,  beginning  with  the  number 
of  shillings  we  have  brought  from  the  pence  column.  Having 
thus  ascertained  the  total  number  of  shillings,  we  find  how 
many  pounds  there  are  in  that  number  of  shillings  by  dividing 
by  20,  there  being  20  shillings  in  the  pound  sterling;  and  after 
having  found  this  number  of  pounds,  we  carry  it  to  the  pounds 
column,  and  the  2  shillings  which  we  find  remaining  we  write 
under  the  shillings  column.  We  then  proceed  to  add  the  pounds 
column,  beginning  with  the  number  of  pounds  in  shillings  which 
we  have  carried  from  the  shillings  column. 

In  adding  up  cwts.,  quarters,  and  pounds,  the  mode  of  pro- 
cedure is  precisely  the  same,  only  as  there  are  28  Ibs.  in  1  quar- 
ter, 4  quarters  in  1  cwt.,  and  20  cwt.  in  1  ton,  the  divisors  we 
use  at  each  step  must  vary  correspondingly.  This  will  be  plain 
from  the  following  example  : 

Here  we  find  the  sum  to  be  20  cwt.  3  qrs.  and  17  Ibs.,  or  1 
ton  0  cwt.  3  qrs.  and  17  Ibs. ;  for,  after  adding  the  first  column, 
and  dividing  the  sum  by  28,  we  have  17  left,  and  after  add- 


60  ARITHMETIC    OF   THE    STEAM-ENGINE. 

cwt.  qr.  Ibs.     ing  the  second  or  quarters  column  with  the 

3     3    12     addition  of  the  number  of  quarters  in  Ibs.  that 

2     318     we  kave  carried  over  from  the  Ibs.  column,  we 

6219     divide  the  number  so  obtained  by  four  to  obtain 

2     0     0     the  number  of  cwts.  there  are  in  all  these  quar- 

1  ton  0    3    17     ^ers<     ^e  carry  the  cwt.  so  obtained  to  the 

:==     cwts.  column,  and  write  beneath  the  quarters 

column  the  3  quarters  which  we  find  are  left.     Proceeding  in 

the  same  way  with  the  cwts.  column,  we  find  its  sum  to  be  20 

cwts.  or  1  ton ;  and  the  total  quantity  to  be  1  ton  0  cwt.  3  qrs.  17 

Ibs.,  as  stated  above. 

Subtraction  of  compound  quantities. — When  we  wish  to 
subtract  one  compound  quantity  from  another,  we  write  the  less 
under  the  greater,  so  that  the  terms  of  the  same  kind  may  be  in 
the  same  column,  as  in  the  case  of  addition.  We  then  subtract 
the  right-hand  term  of  the  lower  line  from  that  of  the  upper,  if 
possible.  But  if  this  cannot  be  done,  we  must  transform  a  unit 
of  the  next  higher  term  into  its  equivalent  number  of  units  of 
the  first  term,  and  then  performing  the  subtraction,  we  write 
the  difference  under  the  first  column,  and  we  increase  by  1  the 
next  term  to  be  subtracted  to  compensate  for  the  unit  previously 
borrowed.  In  algebra,  the  usual  process  of  subtraction  is  to 
change  the  signs  of  the  lower  line,  and  then  to  proceed  as  in 
addition. 

If  we  had  to  take  27Z.  8s.  4$d.  from  34Z.  17s.  9|<Z.,  we  should 
write  down  the  greater  sum  first  and  the  less  under  it,  so  that 
£34  17  9f  pounds  should  fall  under  pounds,  shillings  under 
£27  8  4£  shillings,  and  pence  under  pence.  Taking  %d.  from 
~~Z  ~  37  \&.  we  have  \d.  over,  which  we  write  down,  and 
then  taking  4d.  from  9d.  we  have  5d.  over,  which 


we  also  write  below  the  column  of  pence.  Next  taking  8s. 
from  17s.  we  have  9*.  left,  and  taking  7Z.  from  14Z.  we  have  7£., 
and  carrying  1  to  the  2  appearing  in  the  next  place  we  have  3 
from  3,  which  leaves  nothing.  The  difference,  therefore,  be- 
tween 34Z.  17s.  9f  d.  and  27Z.  8s.  4%d.  is  7Z.  9s.  5%d.  If  we  had  to 
subtract  22Z.  18s.  llf<Z.  from  23£  6.  0£<Z.,  we  should  proceed 
thus : — 


COMPUTATION    OF    COMPOUND    QUANTITIES.  61 

£'23    6    04-  Here  taking  %d.  from  \d.  we  have  to  borrow 

£22  18  11 J  Id.  or  4  farthings  from  the  next  term,  and  we  have 

~~~~ ~    ~  then  6  farthings  to  be  subtracted  from,  and  ftZ. 

-  subtracted  from  £ d.  leaves  £ d.     In  the  next  term 


we  have  11<Z.,  which  must  be  increased  to  12<Z.  on  account  of 
the  penny  before  borrowed ;  and  as  we  have  no  pence  to  sub- 
tract from  we  must  borrow  la.  from  the  next  term,  and  change 
it  into  12  pence,  and  12  pence  taken  from  12  pence  leaves  noth- 
ing. In  the  next  term  of  shillings  we  have  18,  which  must  be 
increased  to  19  in  consequence  of  the  previous  borrowing  of  Is. 
to  carry  to  the  column  of  pence,  and  19s.  taken  from  II.  6s.  or 
26s.  leaves  Is.  In  the  next  term  the  2  has  to  be  increased  to  3 
to  make  up  for  the  11.  imported  into  the  column  of  shillings, 
and  23  taken  from  23  leaves  nothing.  The  difference  between 
these  two  sums  is  consequently  Is.  Of  d. 

If  we  have  to  take  5  tons  12  cwt.  3  qrs.  27i  Ibs.  from  93 
tons'  8  cwt.  1  qr.  6  Ibs.,  we  proceed  as  follows : 
tons  cwt  qr  ibs  Here  %  lb.  taken  from  1  Ib.  leaves  J  lb.,  and 

93    8     1     6       28  Ibs.  taken  from  1  qr.  and  6  Ibs.  or  34  Ibs., 
2H     leaves  6  Ibs.    Then  4  qrs.  taken  from  1  cwt.  and 
87  15    1    6£     1  qr.  or  5  qrs.  leaves  1  qr. ;  and  13  cwt.  taken 
:    from  1  ton  and  8  cwt.  or  28  cwt.  leaves  15  cwt. 
Lastly,  6  tons  taken  from  93  tons  leaves  87  tons. 

If  we  wish  to  subtract  6—2+4  from  9—3  +  2,  we  may  either 
perform  the  subtraction  by  first  adding  the  quantities  together, 
and  then  subtracting  the  sum  of  the  one  from  that  of  the  other, 
or  we  may  change  the  signs  of  the  quantity  to  be  subtracted, 
and  then  add  all  together,  which  will  give  the  same  result. 
Thus  6— 2  =  4,  and  4+4=8.  So  also  9— 3  ==6,  and  6+2  =  8. 
Subtracting  now  one  sum  from  the  other,  we  get  8—8  =  0. 
But  if  we  change  the  signs  of  6—2+4,  and  add  it  to  9—3  +  2, 
we  have  9—3  +  2—6  +  2—4=0. 

Multiplication  of  compound  quantities. — When  we  wish  to 
perform  the  multiplication  of  any  compound  number,  such  as 
pounds,  shillings,  or  pence,  or  hundredweights,  quarters,  and 
pounds,  we  set  the  multiplier  under  the  right-hand  term  of  the 
multiplicand,  multiply  that  term  by  it,  and  find  what  number 


62  ARITHMETIC    OF   THE    STEAM-ENGIHE. 

of  times  one  of  the  next  higher  term  is  contained  in  the  prod- 
uct, which  number  is  to  be  carried  to  the  next  term,  while  the 
remainder,  if  any,  is  to  be  written  under  the  right  hand  or  low- 
est term.     We  must  then  multiply  the  next  term  in  like  man- 
ner, and  so  until  the  whole  have  been  multiplied.    Thus  if  we 
had  to  multiply  23Z.  13*.  5d.  by  4,  we  should  proceed  as  follows  : 
Here  we  first  multiply  the  pence,  and  4  times  5 
4    pence  is  20  pence,  which  is  Is.  8d. ;  and  so  we  put 

down  8  and  carry  1.     In  the  shillings  term  we  say 

£94  13  8  4  tunes  3  are  12,  and  with  the  addition  of  the  1 
shilling  brought  over  from  the  pence  term,  the  12 
becomes  13.  Then  4  times  10  is  40  shillings,  which  make  just  2 
pounds,  so  we  carry  the  2  pounds  to  the  pounds  place,  leaving 
the  13  previously  obtained  in  the  shillings  place.  Proceeding  to 
the  pounds,  we  say  4  times  3  are  12  and  2  are  14,  and  4  times  2 
are  8  and  1  are  9.  Hence  the  product  is  94Z.  18s.  8d.,  which 
sum  would  also  be  obtained  by  writing  down  23Z.  13s.  5d.  four 
times  under  one  another,  and  ascertaining  their  sum  by  addi- 
tion. 

When  the  multiplier  is  large,  but  is  composed  of  two  or  more 
factors,  we  may,  instead  of  multiplying  by  the  number,  multiply 
successively  by  its  factors.  Thus  if  we  have  such  a  sum  as 
£23  lls.  4%d.  to  multiply  by  36,  then  as  36  is  a  number  repre- 
sented by  the  factors  6  x  6,  4  x  9  or  3  x  12,  we  shall  obtain 
the  same  result  by  multiplying  by  any  set  of  these  factors  as  by 
multiplying  by  the  36  direct.  Thus — 

£23  11    4J  £23  11    4|  £23  11    4f 

643 


141     8    4£  94     5    7  70  14    2J- 

6  9  12 


£848  10    3  £848   10    3  £848  10    8 


In  like  manner  if  we  had  to  multiply  the  sum  £17  3s.  Q\d. 
by  140,  then  as  140  is  made  up  of  the  factors  7  x  20,  or 
4  x  5  x  7,  "we  may  multiply  by  these  numbers  instead  of  the 


COMPUTATION   OF   COMPOUND    QUANTITIES.  63 

£17    3    0^     140.     In  cases  however  in  which  the  multiplier 
4       cannot  be  broken  up  into  factors,  we  must  mul- 


68  12    2       tiply  each  term  by  it  consecutively.     Thus  if 

5     '£23  11s.  4$d.  be  multiplied  by  37,  we  have  first 

3  farthings  multiplied  by  37,  which  gives  111 

7       farthings  or  27  pence  and  3  farthings.     Writing 

down  the  3  farthings  and  carrying  the  27  pence, 

5  10  -we  have  37  times  4  pence  or  148  pence,  and  add- 
ing the  27  pence  we  have  175  pence,  which  as 
there  are  12  pence  in  the  shilling  we  divide  by  12  and  get  14 
shillings  and  7  pence.  "We  set  down^  the  7  in  the  pence  place 
and  carry  the  14  to  the  shillings  place,  and  we  thus  proceed 
through  all  the  terms  until  the  multiplication  is  completed.  The 
same  mode  of  procedure  is  adopted  if,  instead  of  pounds,  shil- 
lings, and  pence,  we  have  hundredweights,  quarters,  and  pounds 
or  any  other  quantities  whatever. 

Division  of  compound  quantities. — In  the  arithmetical  divi- 
sion of  compound  quantities,  we  set  the  divisor  in  a  loop  to  the 
left  of  the  dividend  and  divide  the  left-hand  term  by  it,  setting 
the  quotient  under  that  term.  If  there  is  any  remainder  we  re- 
duce it  to  the  next  lower  denomination,  adding  to  it  that  term, 
if  any,  of  the  dividend  which  is  of  this  lower  denomination. 
"We  then  divide  the  result  by  the  divisor  and  so  on,  until  all  the 
terms  have  been  divided.  Thus  if  we  had  to  divide  £38  6s.  8$d. 
by  3,  we  should  proceed  as  follows : — 

£    g     j  Here  we  find  that  3  is  contained  in  3  once,  and  in 

3)38    6    8 ^    8, 2  times  and  2  over.    But  2  pounds  are  40  shillings, 

• -     and  6  are  46  shillings,  and  46  divided  by  3  gives  15 

and  1  over,  which  1  shilling  is  equal  to  12  pence, 
and  adding  to  this  the  8  pence  in  the  dividend,  we  have  20  pence 
to  be  divided  by  3.  Now  20  divided  by  3  gives  6  and  2  over, 
which  2  pence  are  8  farthings,  and  adding  thereto  the  1  farthing 
in  the  dividend,  we  have  9  farthings  to  divide  by  3,  or  3  far- 
things. It  is  clear  that  £12  15s.  6£<Z.  multiplied  by  3  will  again 
give  the  £38  6s.  8$d.  of  the  dividend. 

If  we  have  to  divide  a  number  by  10,  we  may  accomplish  the 
division  by  pointing  off  one  figure  as  a  decimal,  if  by  100  we  point 


04  ARITHMETIC    OF    THE    STEAM-ENGINE. 

off  two  figures,  if  by  1000  three  figures,  and  so  on.     Thus  if  we 

have  to  divide  £2315  14s.  Id.  by  100,  we  may  proceed  as  follows : 

Here  we  point  off  two  figures  of  the  highest 

23'15   14  7     term  as  decimals,  which  leaves  £23.    We  next  mul- 

20  tiply  the  residual  decimal  by  20  to  reduce  it  to 

~TTT  shillings,  bringing  down  the  14  shillings  in  the 

12  dividend,  and  we  obtain  3  shillings  and  '14  of  a 

'  shilling,  which  fraction  we  multiply  by  12  to  bring 

4  it  to  pence,  and  we  bring  down  thereto  the  7  pence 

in  the  dividend.    "We  obtain  as  a  product  1'75  pence, 

and  multiplying  in  like  manner  '74  by  4  to  bring  it  to 

farthings,  we  obtain  3  farthings,  making  the  total  quotient  £23 

3«.  If  d.     This  sum  multiplied  by  100  will  make  £2315  14*.  7d. 

When  the  divisor  is  large  but  may  be  broken  up  into  factors,  we 

may  divide  separately  by  those  factors.    Thus  if  we  wish  to  divide 

£3762  3s.  6d.  by  24,  then  as  24=  4  x  6  or  3  x  8  or  2  x  12,  we 

may  divide  the  sum  by  any  pair  of  factors  instead  of  by  the  24. 

£        s.   d.  £        s.   d.  £        s.    d. 

4)3762     3     6  3)3762     3     6  2)3762     3     6 

6)940  10  10£  8)1254     1     2  12)1881     1     9 

£156  15     If  £156  15     If  £156  15     If 


When  the  number  cannot  be  broken  up  into  factors  we  must 
proceed  by  the  method  of  long  division.     Thus  if  we  had  to 
divide  £3715  18s.  9d.  by  47  we  should  proceed  as  follows : — 
£      s.    d.  Here  we  find  first  how  often  47  will 

8   9(79   1   3     go  ™  371,  and  we  find  it  wiU  be  7  times, 
when  we  write  the  7  in  the  quotient  and 
multiply  the  divisor  by  it,  setting  the 
—  product  under  the  first  three  figures  of 

2  the  dividend.     Subtracting  now  the  329 

from  the  371,  we  find  that  the  remainder 

68(1  is  42,  and  we  bring  down  the  next  figure 

of  the  dividend  and  find  how  often  47  is 
11  contained  in  425.     We  find  that  it  is  9 

times,  which  completes  the  division  of 
141(3  the  pounds.     The  2  pounds  remaining 

we  next  multiply  by  20  to  bring  them  to 


NATURE    OF   AN    INFINITE    SERIES.  65 

shillings,  adding  the  18  shillings  of  the  dividend,  which  together 
make  58  shillings,  the  47th  part  of  which  is  1  shilling  and  J|ths 
over.  Multiplying  this  by  12  to  bring  it  to  pence,  and  dividing 
by  47,  we  get  3  pence,  which  completes  the  operation. 

In  cases  where  we  have  to  divide  a  compound  quantity  by 
another  of  the  same  kind,  such  as  money  by  money  or  weights 
by  weights,  the  requirement  is  equivalent  to  that  of  finding 
what  number  of  times  the  one  amount  is  comprehended  in  the 
other.  We  cannot  of  course  divide  a  quantity  by  another  of  a 
different  kind,  as  money  by  weight,  nor  can  we  multiply  money 
by  money  or  weight  by  weight.  If  we  are  required  to  divide 
such  a  sum  as  £3  7s.  §d.  by  16s.  10|<Z.,  we  reduce  both  the  num- 
bers to  the  lowest  denomination  appearing  in  either,  which  in 
this  case  is  half  pence,  and  we  then  divide  the  greater  number 
by  the  less.  Now  £3  7s.  6d.  =  1620  half  pence  and  16s.  10i<Z. 
=  405  half  pence,  and  1620  -f-  405  =4.  So  if  we  had  to  divide 
3  tons  2  cwt.  2  qrs.  21  Ibs.  by  2  qrs.  7  Ibs.,  then  as  the  first 
amount  is  equal  to  6993  Ibs.  and  the  second  to  63,  the  question 
becomes  one  of  dividing  6993  by  63,  which  we  find  gives  111. 
It  follows  consequently  that  2  qrs.  7  Ibs.  multiplied  by  111  =3 
tons  2  cwt.  1  qr.  21  Ibs. 

As  a  square  foot  contains  144  square  inches,  we  must,  in  as- 
certaining the  number  of  square  feet  in  any  given  number  of 
square  inches,  divide  by  the  number  144,  and  as  a  cubic  foot 
contains  1728  cubic  inches,  we  must,  in  ascertaining  what  num- 
ber of  cubic  feet  there  are  in  any  number  of  cubic  inches,  divide 
by  the  number  1728.  So  also  there  are  nine  square  feet  in  a 
square  yard,  and  27  cubic  feet  in  a  cubic  yard.  A  cubic  foot 
contains  very  nearly  2200  cylindric  inches  or  solid  cylinders  1 
inch  in  diameter  and  1  inch  high ;  3300  spherical  inches  or  balls 
1  inch  diameter ;  and  6600  conical  inches  or  cones  1  inch  diam- 
eter and  1  inch  high. 

ON  THE   RESOLUTION   OF  FRACTIONS  INTO  INFINITE  SERIES. 

We  have  already  explained  that  in  decimal  fractions  the  de- 
crease at  every  successive  figure  is  ten  times,  just  as  in  common 


66  ARITHMETIC    OF   THE    STEAM-ENGINE. 

numbers  the  increase  at  every  successive  number  is  ten  times. 
Thus  the  number  666  means  600  +  60+6,  so  that  the  first 
figure  by  virtue  of  its  position  alone  is  ten  times  greater  than 
the  second,  and  the  second  by  virtue  of  its  position  alone  is  ten 
times  greater  than  the  third.  Precisely  the  same  law  holds 
when  we  descend  below  unity,  as  we  do  in  every  case  in  which 
the  decimal  point  is  introduced,  as  the  meaning  of  the  decimal 
point  is,  that  all  the  numbers  to  the  right  of  it  are  less  than 
unity,  and  that  they  diminish  ten  times  at  each  successive  figure, 
just  as  ordinary  numbers  do.  The  expression  666*666  therefore 
means  six  hundred  and  sixty-six  with  the  addition  of  6  tenths, 
six  hundredths,  and  six  thousandths,  or,  what  is  the  same  thing, 
of  666  thousandths.  The  expression  might  therefore  be  written 
666 +-5-^+^+1-5^,5.  or  eee/^V  Every  decimal  fraction  may 
consequently  be  considered  as  a  vulgar  fraction,  with  a  denom- 
inator of  10  or  100  or  1000  understood,  according  to  the  position 
of  the  decimal.  Thus  '1  is  equivalent  to  T*ff,  '01  is  equivalent 
to  yitf,  and  '001  is  equivalent  to  T^Vff-  Now  the  fraction  ^  is  1 
divided  by  3,  and  if  we  perform  the  division  we  shall  have 

3)1-00000 


•33333,  &c., 

and  so  on  to  infinity.  The  vulgar  fraction  £  is  consequently 
equal  to  the  infinite  series  -83333,  &c.,  which,  at  each  successive 
term  to  which  it  is  carried,  becomes  more  nearly  equal  to  the 
fraction  of  •&,  but  never  becomes  exactly  equal  thereto.  Any 
vulgar  fraction  may  be  at  once  converted  into  its  equivalent 
decimal  by  dividing  the  numerator  by  the  denominator,  adding 
as  many  ciphers  to  the  numerator  as  may  be  necessary  to  enable 
the  division  to  be  carried  on.  But  some  of  the  divisions  thus 
performed,  it  will  be  found,  may  be  carried  on  for  ever,  and  such 
a  series  of  numbers  is  termed  an  infinite  series.  As  a  visible 
exemplification  of  the  continual  approach  of  two  quantities  to 
one  another  without  ever  becoming  equal,  we  may  take  the  fol- 
lowing example : 

Here  we  have  a  line  A  B  which  we  may  divide  into  any  num- 


ARITHMETICAL    EXAMPLES. 


67 


ber  of  equal  parts,  and  we  draw  the  line  A  o  at  right  angles 
with  A  B  :  at  o  we  draw  another  short  line  ac  parallel  to  A  B,  and 
we  set  off  the  distance  ca  equal  to  Al.  If  now  we  draw  the 
diagonal  line  la  we  shall  cut  off  the  half  of  A  c,  or  shall  bisect 
it  in  the  point  x,  and  by  drawing  the  lines  2a,  3a,  4a,  5a,  &c., 


we  cut  off  successive  portions  of  xc,  and  therefore  continually 
diminish  it.  But  we  never  can  cut  it  all  off,  however  extended 
we  may  make  the  line  A  B,  and  however  numerous  the  addi- 
tional portions  cut  off  may  be.  The  quantity  xc  becomes  more 
and  more  nearly  equal  to  ZA,  the  greater  the  length  of  the  line 
A  B,  and  the  more  numerous  the  fractional  quantities  successively 
cut  off.  But  no  extension  of  the  operation  short  of  infinity 
could  make  the  portion's  cut  off  from  xo  equal  to  »A. 


ARITHMETICAL  EXAMPLES. 

Having  now  illustrated  with  adequate  fulness  of  detail  the 
elementary  principles  of  engineering  arithmetic,  it  is  only  neces- 
sary that  we  should  add  some  examples  of  the  method  of  per- 
forming such  computations  as  are  most  likely  to  be  required  in 
practice. 

KEDUCTION. — This  is  the  name  given  to  the  process  of  con- 
verting a  quantity  expressed  in  one  denomination  into  an  equiv- 
alent quantity  expressed  in  another  denomination,  such  as  tons 
expressed  in  ounces,  or  miles  in  yards. 

Example  1. — Eeduce  151.  7«.  Ofd.  to  farthings. 


68  ARITHMETIC    OF   THE    STE AIM-ENGINE. 

15    7    OJ  Here  we  first  multiply  the  pounds  by  20,  there 

being  20  shillings  in  the  pound,  and  we  bring 
307s.  down  the  7  shillings,  making  307  shillings.     We 

12  then  multiply  the  shillings  by  12,  there  being 

~~~  12  pence  in  the  shilling,  and  here  we  have  no 

4  '  pence  to  bring  down.     Finally,  we  multiply  by 

4,  there  being  4  farthings  in  each  penny,  and 


14739/.  Am.  WQ  bring  down  the  3  fartuingSj  making  14,739 
farthings  in  all. 

Example  2. — Eeduce  23  tons  to  pounds  avoirdupois. 

By  a  reference  to  a  table  of  weights  and  measures,  we  find 
that  there  are  2,240  pounds  in  the  ton ;  23  times  2,240,  there- 
fore, or  51,520  Ibs.,  is  the  answer  required. 

Example  3. — Eeduce  100  square  yards  to  square  inches. 
Here,  as  each  square  yard  contains  9  square  feet,  and  each  square 
foot  144  square  inches,  there  will  be  9  times  144  or  1,296  square 
inches  in  each  square  yard,  and  100  times  this; or  129,600  square 
inches,  in  100  square  yards.  It  may  be  well  here  to  remark  that 
100  square  yards  is  a  very  different  quantity  from  100  yards 
square,  which  would,  in  fact,  contain  an  area  of  10,000  square 
yards. 

Example  4. — Eeduce  7  cubic  yards,  20  cubic  feet,  to  cubic 
inches.  As  there  are  27  cubic  feet  hi  a  cubic  yard,  there  will 
be  27  times  7,  or  189  cubic  feet  in  7  cubic  yards,  to  which  add- 
ing 20,  we  have  209  cubic  feet  in  all;  and  as  there  are  1,728 
cubic  inches  in  a  cubic  foot,  we  have  1,728  times  209,  or  361,152 
cubic  inches  as  the  answer  required. 

Quantities  are  brought  to  a  higher  denomination  by  the  re- 
verse of  the  process  indicated  above,  that  is,  by  dividing,  instead 
of  multiplying.  Thus,  by  dividing  by  4,  12,  and  20,  it  will  be 
found  that  14,739  farthings  are  equal  to  15 1.  7s.  Ofd. ;  by  divid- 
ing 51,520  Ibs.  by  2,240,  that  the  quotient  is  equal  to  23  tons ; 
and  by  dividing  129,600  square  incnes  by  144,  and  then  by  9, 
that  the  result  is  100  square  yards.  So  also  by  dividing  by 
1,728,  it  will  be  found  that  361,152  cubic  inches  are  equal  to 
209  cubic  feet,  and  dividing  again  by  27,  we  find  the  answer  to 
be  7  cubic  yards  and  20  cubic  feet. 


EXAMPLES    OF    PRACTICAL    COMPUTATIONS.  69 


N  OF  SURFACES  AXD  SOLIDS.  —  The  area  of  a  rec- 
tangular surface  is  obtained  by  multiplying  the  length  by  the 
breadth.  The  area  of  a  circle  in  circular  inches  is  obtained  by 
multiplying  the  diameter  by  itself;  and  the  area  of  a  circle  in 
square  inches  is  obtained  by  multiplying  the  diameter  by  itself, 
and  by  the  decimal  "7854.  The  circumference  of  a  circle  is 
3-1416  times  its  diameter.  The  capacity  of  a  rectangular  solid 
is  obtained  by  multiplying  together  its  length,  depth,  and  thick- 
ness ;  and  the  capacity  of  a  cylinder  in  cubic  feet  or  inches  is 
obtained  by  multiplying  the  area  of  its  cross  section  or  mouth, 
expressed  in  square  feet  or  inches,  by  its  depth  in  feet  or  inches. 

Example  1.  —  What  is  the  quantity  of  felt  required  to  cover 
the  side  of  a  marine  boiler  that  is  17  feet  8  inches  long,  and  3 
yards  high  ? 

Here  we  first  reduce  the  measurements  to  inches,  and  as  17 
ft.  8  in.  is  equal  to  212  inches,  and  as  3  yards  or  9  feet  is  equal 
to  108  inches,  we  have  an  area  represented  by  212  multiplied 
by  108  inches,  or  22,896  square  inches.  Now,  as  there  are  144 
square  inches  in  each  .square  foot,  we  shall,  by  dividing  22,896 
by  144,  find  that  the  area  is  159  square  feet,  and  dividing  this 
by  9  to  bring  the  quantity  into  square  yards,  we  find  that  the 
area  is  17  square  yards  and  6  square  feet  over. 

Since  the  area  is  obtained  by  multiplying  the  length  by  the 
breadth,  it  will  follow  that  if  we  divide  the  area  by  the  length 
we  shall  get  the  breadth,  and  if  we  divide  the  area  by  the 
breadth  we  shall  get  the  length. 

Example  2.  —  What  is  the  weight  required  to  be  placed  on 
top  of  a  safety-valve  4  inches  diameter,  to  keep  it  down  until 
the  steam  attains  a  pressure  of  20  Ibs.  on  each  square  inch  ? 

Here  4x4=16  circular  inches,  and  16  x  "7854=  12-566 
square  inches,  which  x  20  the  pressure  on  each  square  inch  = 
251-32  Ibs. 

Example  2.  —  The  engine  of  the  steamer  'Arrogant'  is  a 
trunk  engine,  in  which  the  piston  rod  is  widened  into  a  hollow 
trunk  or  pipe  24  inches  diameter,  which  correspondingly  re- 
duces the  effective  area  of  the  piston.  As  the  cylinder  is  60 
inches  diameter,  reduced  by  a  circle  24  inches  diameter,  what 


70  ARITHMETIC    OF   THE    STEAM-ENGINE. 

will  be  the  diameter  of  a  common  cylinder  to  have  an  equal 
area? 

Here  602  x  '7854  =•  2827'44  square  inches,  and  243x*7854  = 
452-39  square  inches,  and  2827'44  diminished  by  452*39  —  2375-05 
square  inches.  This  is  as  nearly  as  possible  the  area  of  a  cylin- 
der 55  inches  in  diameter,  which  is  2375-83  square  inches. 

Example^. — The  steamer  'Black  Prince'  has  two  direct- 
acting  trunk  engines,  with  cylinders  equal  to  104%  inches  diam- 
eter, and  the  length  of  the  stroke  is  4  feet.  The  engines  make 
55  revolutions  per  minute.  "What  will  be  the  number  of  cubic 
feet  of  steam  required  per  hour  to  fill  the  cylinder  ? 

Here  the  diameter  being  104£  inches,  the  area  of  each  cylin- 
der will  be  -7854  times  104J  squared,  or  it  will  be  8835*7  square 
inches,  or  61 -3  square  feet.  As  the  piston  travels  backwards 
and  forwards  at  each  revolution,  it  will  pass  through  8  feet  dur- 
ing each  revolution ;  and  the  volume  of  steam  required  by  each 
cylinder  in  each  revolution  will  be  8  times  61-3,  or  490-4  cubic 
feet.  As  there  are  two  engines,  the  total  volume  of  steam  re- 
quired in  each  revolution  will  be  twice  490*4,  or  it  will  be  980-8 
cubic  feet ;  and  as  there  are  55  strokes  in  each  minute,  the 
expenditure  per  minute  will  be  55  times  980-8,  or  53,944  cubic 
feet.  The  expenditure  per  hour  will,  of  course,  be  60  times 
this,  or  3,236,640  cubic  feet.  In  all  modern  engines  the  steam 
is  not  allowed  to  enter  the  cylinder  from  the  boiler  during  the 
whole  stroke;  and  the  expenditure  of  steam  will  be  less  the 
sooner  it  is  cut  off  or  prevented  from  entering  the  cylinder. 
But  the  cylinder,  nevertheless,  will  still  be  filled  with  steam, 
though  of  a  less  tension,  than  if  the  supply  from  the  boiler  had 
not  been  interrupted ;  and  the  space  traversed  by  the  piston  will 
always  be  a  correct  measure  of  the  steam  consumed,  taking  that 
steam  at  the  pressure  it  has  at  the  end  of  the  stroke. 

Example  4. — The  '  Black  Prince '  has  an  area  of  immersed 
midship  section  of  1,270  square  feet ;  or,  in  other  words,  if  the 
vessel  were  cut  across  in  the  middle,  the  area  of  that  part  below 
the  water  would  be  1,270  square  feet.  The  diameter  of  the 
screw  is  24  feet  6  inches,  the  nominal  power  is  1,250,  and  the 
indicated  power  5,772  horses.  What  is  the  ratio,  or  proportion, 


EXAMPLES    OF   PRACTICAL   COMPUTATIONS.  71 

of  the  area  of  midship  section  to  the  area  of  the  circle  in  which 
the  screw  revolves  ?  and  what  is  the  ratio  of  the  immersed  mid- 
ship section  to  the  indicated  power? 

Here  the  diameter  of  the  screw  being  24£  feet,  the  area  of 
the  circle  in  which  it  revolves  will  be  471  "436  square  feet,  and 
1,270  divided  by  471-436  being  2-69,  it  follows  that  the  ratio  of 
immersed  midship  section  to  screw's  disc  is  2'69  to  1.  So,  in 
like  manner,  the  indicated  power  5,772,  divided  by  1,250,  gives 
a  ratio  of  indicated  power  to  immersed  midship  section  of  4'54 
to  1.  With  these  proportions  the  speed  was  at  the  rate  of  nearly 
15  knots  per  hour,  so  that  to  ensure  such  a  speed  in  a  vessel 
like  the  '  Black  Prince,'  it  is  necessary  that  there  should  be  4^ 
or  5  indicated  horse-power  for  each  square  foot  of  immersed 
midship  section  of  the  hull. 

Example  5. — If  it  were  desired  to  encircle  the  screw  of  the 
'  Black  Prince '  with  a  sheet-iron  hoop,  what  length  of  hoop 
would  be  required  for  the  purpose  ? 

The  diameter  of  the  screw  being  24i  feet,  the  circumference 
of  the  circle  in  which  it  revolves  will  be  3'1416  times  24i,  or  it 
will  be  76-969  feet. 

Example  6. — A  single  acting  feed  pump  has  a  ram  of  2£  inches 
diameter  and  18  inches  stroke,  and  makes  50  strokes  per  minute. 
How  much  water  ought  it  to  send  into  the  boiler  every  hour  ? 

Here  the  area  of  the  ram  will  be  4'9  square  inches,  and  the 
stroke  being  18  inches,  18  times  4*9  or  88'2  cubic  inches  will  be 
expelled  at  every  stroke,  supposing  that  there  is  no  loss  by  leak- 
age or  otherwise.  As  there  are  50  strokes  made  in  the  minute, 
the  discharge  per  minute  will  be  50  times  88-2,  or  4,410  cubic 
inches;  and  there  will  be  60  times  this,  or  264,600  cubic  inches 
discharged  ia  the  hour.  As  there  are  1,728  cubic  inches  in  the 
cubic  foot,  we  get  the  hourly  discharge  in  cubic  feet  by  dividing 
264,600  by  1,728,  and  we  shall  find  the  discharge  to  be  153-125 
cubic  feet.  A  cubic  inch  of  water  will  make  about  a  cubic  foot 
of  steam,  of  the  same  pressure  as  the  atmosphere. 

Example  7. — A  cubic  foot  of  water  weighs  1,000  ounces. 
What  will  be  the  weight  of  water  in  a  vessel  which  is  filled  to 
the  brim,  and  which  measures  a  yard  each  way  ? 


72  ARITHMETIC    OF   THE    STEAM-ENGINE. 

As  tiiere  are  27  cubic  feet  in  a  cubic  yard,  the  weight  re- 
quired will  be  27,000  ounces,  which,  divided  by  16,  the  number 
of  ounces  in  a  pound,  gives  1,687  Ibs.  and  8  oz.,  and  dividing 
again  by  112,  the  number  of  Ibs.  in  each  cwt.,  we  get  15  cwt. 
7  Ibs.  8  oz. 

Example  8. — Two  steamers  being  started  together  on  a  race, 
it  was  found  that  the  faster  went  5  feet  ahead  of  the  other  in 
each  55  yards :  how  much  will  she  have  gained  in  half  a  mile  ? 

As  a  mile  is  1,760  yards,  half  a  mile  is  880  yards,  and  there 
are  16  times  55  yards,  therefore,  in  half  a  mile.  As  in  each  55 
yards  5  feet  are  gained,  there  will  be  16  tunes  5  feet,  or  80  feet 
gained  in  the  half  mile,  or  26  yards  2  feet. 

Example  9. — The  'Warrior,'  a  steamer  of  6,039  tons  burden, 
and  1,250  nominal  horse-power,  attained  a  speed  on  trial  of 
14-356  knots  per  hour,  the  engines  exerting  an  actual  power  of 
5,469  horses.  The  screw  was  24J  feet  diameter,  and  30  feet 
pitch,  or,  in  other  words,  the  twist  of  the  blades  was  such  that  it 
would  advance  30  feet  at  each  revolution,  if  the  advance  were 
made  without  any  resistance.  The  engines  made  54-25  revolu- 
tions per  minute,  and  if  the  screw  advanced  30  feet  in  each  revo- 
lution, it  would  advance  1627*5  per  minute,  or  16-061  knots  per 
hour.  In  reality,  however,  the  screw  only  advanced  through 
the  same  distance  as  the  ship,  namely,  14-356  knots  per  hour. 
The  actual  advance,  therefore,  was  less  than  the  theoretical  ad- 
vance by  1-705  knots  per  hour,  which  difference  is  called  the 
slip  of  the  screw;  for  1'705  added  to  14-356  makes  16-061, 
which  would  be  the  speed  of  the  vessel  at  this  speed  of  the  screw 
if  there  was  no  slip. 

Example  10. — What  is  the  diameter  of  a  piston  of  which  the 
area  is  2827'44  square  inches  ? 

Here  2827'44,  divided  by  '7854,=  3600,  the  square  root  of 
which  is  60.  This  is  the  diameter  required. 

Example  11. — A  cubical  vessel  of  water  weighs  5  tons,  ex- 
cluding the  weight  of  the  vessel.  What  is  the  length  of  the 


As  there  are  1,000  ounces  in  a  cubic  foot  of  water,  we  know 
that  there  will  be  the  same  number  of  cubic  feet  in  the  vessel  as 


EXAMPLES    OF   PRACTICAL    COMPUTATIONS. 


73 


156 


7500 
936 

8436 


1683 


940800 
5049 


945849 


179-2(5-63 
125 


54-200 


50-616 


3584000 


_2837547 
~746453 


the  number  of  times  1,000  ounces  is  contained  in  5  tons.  Now 
as  there  are  2,240  Ibs.  in  the  ton,  there  will  be  5  times  this,  or 
11,200  Ibs.  in  5  tons,  or  179,200  ounces.  Dividing  this  by  1,000, 
we  have  179-2  cubic  feet  as  the  content  of  the  vessel. 

To  find  the  length  of  the  side  we  must  extract  the  cube  root 

of  179-2.  We  soon  see  that 
the  root  must  lie  between  5 
and  6,  for  the  cube  of  5  is 
125,  and  the  cube  of  6  is  216. 
Taking  5  as  the  next  lowest 
root,  we  set  this  number  as 
the  first  figure  of  the  quotient 
and  subtract  its  cube  as  in 
long  division,  bringing  down 
three  more  figures  at  each 
stage,  and  here  two  of  these  must  be  ciphers. 

We  now  triple  the  root  5,  and  set  down  the  15  to  the  left, 
and  we  multiply  this  triple  number  by  the  first  figure  of  the 
root  making  75,  which  number  we  set  down  between  the  15 
and  the  remainder,  adding  two  ciphers  to  it,  which  make  it 
7500.  We  now  consider  how  often  the  trial  divisor  7500  will 
go  into  the  remainder  54200,  after  making  some  allowance  for 
additions  to  the  divisor,  and  we  find  it  will  be  6  times.  We 
place  the  6  as  the  second  figure  of  the  root,  and  we  also  place  it 
after  the  15.  We  multiply  the  156  by  the  6,  and  place  the  prod- 
uct under  the  7500.  The  resulting  number,  8436,  is  the  first 
true  divisor. 

We  now  bring  down  the  next  period  of  three  figures,  and  as 
there  are  no  figures  remaining  to  be  brought  down,  we  introduce 
three  ciphers.  We  triple  the  last  figure  of  156,  which  gives  168, 
and  we  add  the  square  of  6,  which  is  36,  to  the  sum  of  the  two 
last  lines,  936  and  8436,  making  in  all  9408,  to  which  we  add 
two  ciphers,  making  940800,  and  we  then  see  how  often  this 
sum  is  contained  in  3584000.  We  find  that  it  will  be  3  times, 
and  we  set  down  the  3  as  the  next  figure  of  the  root,  and  also 
after  the  1C8,  making  1683,  and  we  add  three  times  this  to  the 
940,800,  making  945849,  which  is  the  second  real  divisor.  We 


74  ARITHMETIC   OF   THE   STEAM-ENGINE. 

now  multiply  this  divisor  by  the  last  figure  of  the  quotient,  and 
subtract  the  product  as  in  long  division,  leaving  as  a  remainder 
746453,  to  which,  if  we  wished  to  carry  the  answer  to  another 
place  of  decimals,  we  should  annex  three  ciphers,  and  proceed 
as  before.  For  all  ordinary  purposes,  however,  an  extraction  to 
the  second  place  of  decimals  is  sufficient,  and  if  we  cube  5-63,  we 
shall  find  the  resulting  number  to  be  178-453547,  being  a  little 
less  than  179-2. 

Example  12. — The  density  or  specific  gravity  of  mercury  is 
13*59  times  greater  than  that  of  water,  and  the  specific  gravity 
of  water  is  773-29  tunes  greater  than  that  of  air  of  the  usual  at- 
mospheric pressure.  What  will  be  the  height  of  a  column  of 
water  that  will  balance  the  usual  barometric  pressure  of  30 
inches  of  mercury,  and  what  also  will  be  the  height  of  a  column 
of  air  of  uniform  density  that  will  be  required  to  balance  that 
pressure  t 

Here  the  mercury  being  13-59  times  more  dense  than  the 
water,  or  in  other  words,  the  water  being  13-59  tunes  more  light 
than  the  mercury,  it  will  be  necessary  that  the  height  of  the 
column  of  water  should  be  13-59  times  greater  than  that  of  the 
column  of  mercury,  in  order  to  balance  the  pressure.  If,  there- 
fore, the  column  of  mercury  be  30  inches  high,  the  height  of  the 
balancing  column  of  water  must  be  13-59  times  30  inches,  or 
33-975  feet,  and  the  height  of  the  balancing  column  of  air  must 
be  773-29  tunes  this,  or  26271  '52775  feet.  In  point  of  fact,  the 
height  will  be  a  little  more  than  this,  as  mercury  is  13-59593 
times  heavier  than  water,  whereas,  for  simplicity,  it  has  been 
taken  here  at  only  13-59  times  heavier. 

EQUATIONS. 

"When  one  quantity  is  set  down  as  equal  to  another  quantity 
with  the  sign  of  equality  (  =  )  between  the  two,  the  whole  ex- 
pression is  termed  an  equation.  Thus  1  Ib.  =  16  oz.  is  an  equa- 
tion ;  and  if  we  represent  Ibs.  by  the  letter  A,  and  oz.  by  the 
letter  B,  then  we  shall  have  the  equation  in  the  form  IA  or 
A  =  16s.  It  is  clear  that  the  equality  subsisting  in  such  an  ex- 


NATURE   AND   USES    OF   EQUATIONS.  75 

pression  will  not  be  extinguished  by  any  amount  of  addition, 
subtraction,  multiplication,  division,  or  other  arithmetical  pro- 
cess to  which  it  may  be  subjected,  provided  it  be  simultaneously 
applied  to  both  sides  of  the  equation — just  as  the  equality  of 
weight  shown  by  a  pair  of  scales  between  1  Ib.  and  16  oz.  will 
not  be  altered  if  we  add  an  ounce,  or  pound,  or  any  other  weight 
to  each  scale,  or  subtract  an  ounce,  or  pound,  or  any  other 
weight  from  each  scale.  If  we  add  an  ounce  to  each  scale,  then 
we  shall  have  the  equation  A  +  B=  16s  +  B,  or  if  we  subtract 
an  ounce  from  each  scale,  the  equation  becomes  A— B  =  16s— B, 
both  of  which  expressions  are  obviously  just  as  correct  as  the 
first  one.  We  may,  consequently,  add  any  quantity  to  each  side 
of  an  equation,  or  subtract  any  quantity  from  it  without  altering 
the  value  of  the  expression. 

If  we  have  such  an  expression  as  A  —  B  =  16s  —  B,  and  wish 
thereby  to  know  the  value  of  A,  we  shall  ascertain  it  by  adding 
the  quantity  B  to  each  side  of  the  equation,  which  will  then  be- 
come A  —  B  +  B  =  16s  —  B+B.  Now  A — B  +  B  is  obviously  equal 
to  A,  for  the  value  of  any  quantity  is  not  changed  by  first  sub- 
tracting and  then  adding  any  given  quantity  to  it.  So  likewise 
16B  —  B  +  B  is  obviously  equal  to  16s,  as  the  — B  and  +  B  de- 
stroy one  another.  The  equation  thus  cleared  of  redundant 
figures  becomes  A  =  16s.  as  at  first. 

If  now  we  divide  both  sides  of  the  equation  by  any  number, 
or  mutiply  both  sides  by  any  number,  we  shall  find  the  value  of 
the  expression  to  remain  without  change.  For  example,  if  we 

divide  by  16  we  shall  get  ^  =  B,  or  if  we  multiply  by  2  we  shall 

get  2A  =  32s.  Both  of  these  expressions  are  obviously  as  true 
as  the  first  one,  as  they  amount  to  saying  that  -j*gth  of  a  pound  is 
equal  to  an  ounce,  and  that  2  Ibs.  are  equal  to  32  oz. 

If  we  have  such  an  expression  as  a  +  ft  =-  c,  and  wish  to  know 
the  value  of  <z,  we  subtract  &  from  both  sides  of  the  equation, 
which  we  have  seen  we  can  do  without  error,  whatever  quan- 
tity &  may  be  supposed  to  represent.  Performing  this  subtrac- 
tion we  get  a  +  Z»— &,  or  a  «-  c— J ;  and  if  we  know  the  values 
of  c  and  &,  we  at  once  get  the  valne  of  a.  If  we  know  the 


76  ARITHMETIC    OF   THE    STEAM-ENGINE. 

values  of  a  and  c,  and  wish  to  find  the  value  of  &,  we  shall 
ascertain  it  by  substracting  a  from  each  side  of  the  equation, 
which  will  then  become  5  =  c  —  a.  In  both  of  these  subtrac- 
tions we  may  see  that  we  have  merely  shifted  a  letter  from  one 
side  of  the  equation  to  the  other,  at  the  same  time  changing  its 
sign ;  and  we  hence  deduce  this  general  law  applicable  to  all 
equations,  that  we  may  without  error  transfer  any  quantity  from 
the  one  side  to  the  other,  if  we  at  the  same  time  change  its  sign. 

If  we  have  the  equation  a  =  v,  and  if  we  know  the  values  of 

a  and  5,  but  not  of  x,  then,  to  find  the  value  of  a;,  we  multiply 
both  sides  of  the  equation  by  &,  which  reduces  the  equation  to 
the  form  db  =  x.  If,  then,  a  =  2  and  &  =  4,  it  is  clear  that 
x  =  8.  It  may  be  here  remarked  that  ab  is  the  same  as  a  x  0, 
and  which  is  quite  a  different  expression  from  a  +  0,  the  one 
meaning  a  multiplied  by  &,  and  the  other  a  added  to  0.  So  like- 

db  ,  ab 

wise  -r-  =  a  and  —  =  0. 
b  a 

The  utility  of  such  equations  in  engineering  computations  is 
very  great,  not  merely  as  simplifying  arithmetical  processes,  but 
as  presenting  compendious  expressions  of  important  laws,  both 
easily  remembered  and  easily  recorded.  Thus  it  is  found  that 
in  steam-vessels  the  power  necessary  to  be  put  into  them,  to 
achieve  any  given  speed  with  any  given  form  of  vessel,  and  any 
given  area  of  immersed  midship  section,  varies  as  the  cube  of 
the  speed  required.  If  we  represent  the  indicated  power  by  P, 
the  speed  in  knots  per  hour  by  s,  the  area  in  square  feet,  and 
the  cross  section  below  the  water  line  by  A,  and  if  by  o  we  de- 
note a  certain  multiplier  or  coefficient,  the  value  of  which  varies 
with  the  form  of  the  vessel,  but  is  constant  in  the  same  species 

S^A 

of  vessel,  then  P  =  —  is  an  equation  which  expresses  these  re- 
lations, and  we  can  find  the  value  of  P  from  this  equation  if  we 
know  the  value  of  the  other  quantities,  or  we  can  find  the  value 
of  8,  or  of  A,  or  of  o,  if  we  know  the  values  of  the  other  quan- 
tities in  the  equation.  Thus  if  we  multiply  both  sides  of  the 
equation  by  o,  we  get  PO  ==  83A,  and  if  we  now  divide  by  p  we 


EXAMPLES    OF   EQUATIONS .  77 

S3A. 

get  o  = So  also  if  we  divide  the  equation  PC  =  s3A  by  s3, 

PC 

we  get  the  value  of  A,  as  we  shall  then  have  —  =  A  ;  or  if  we 

S' 
PO 

divide  by  A  we  get  —  =  s3,  and  taking  out  the  cube  root  of  both 

A 

3    / 

/  PU 

sides  we  get  y  —  =  s.    If,  therefore,  we  know  the  indicator 

A 

power  of  a  steamer,  the  immersed  area  of  midship  cross  section, 
and  the  coefficient  proper  for  the  order  of  vessel  to  which  the 
particular  vessel  under  examination  belongs,  we  can  easily  tell 
what  the  speed  will  be,  as  we  have  only  to  multiply  the  indi- 
cator power  in  horses  by  the  coefficient,  and  divide  by  the  sec- 
tional area  in  square  feet,  and  finally  to  extract  the  cube  root  of 
the  quotient,  which  will  give  the  speed  in  knots  per  hour.  The 
coefficients  of  different  vessels  have  been  ascertained  by  experi- 
ment. The  following  are  the  coefficients  of  some  of  the  screw- 
vessels  of  the  navy : — 

'Shannon,'  650;  'Simoom,'  500;  'Windsor  Castle,'  493; 
'Penguin,'  648;  'Plover,'  670;  'Curasoa,'  677;  'Himalaya,' 
695;  'Warrior,'  824;  '  Black  Prince,' 674.  The  coefficient  of 
the  Eoyal  Yacht  '  Fairy '  is  464,  and  the  original  coefficient  of 
the  '  Rattler '  was  676 ;  but  the  performance  has  latterly  fallen 
off,  and  is  not  now  above  500,  or  thereabout.  The  original  co- 
efficient of  the  '  Frankfort,'  a  merchant  screw  steamer,  was  792, 
which  was  about  the  best  performance  at  that  time  attained. 
The  larger  the  coefficient  the  better  is  the  performance. 


CHAPTER  H. 

MECHANICAL  PRINCIPLES  OF  THE  STEAM-ENGINE. 


LAW  OF  CONSERVATION  OF  FORCE. 

THE  fundamental  principle  of  Mechanics,  as  of  Chemistry, 
Physiology,  and  every  department  of  physical  science,  is  that  a 
force  once  in  being  can  never  cease  to  exist,  except  by  its  trans- 
formation into  some  other  equivalent  force,  which,  however, 
does  not  involve  the  annihilation  of  the  force,  as  it  continues  to 
exist  in  another  form.  This  principle,  usually  termed  the  con- 
servation of  force,  and  sometimes  the  conservation  of  energy,  is 
only  now  beginning  to  receive  that  wide  and  distinct  recognition 
which  its  importance  demands ;  and  it  will  be  found  that  the 
clear  apprehension  of  this  pervading  principle  will  greatly  sim- 
plify and  aid  all  our  investigations  in  natural  science.  One  very 
obvious  inference  from  the  principle  is  that  we  cannot  manufac- 
ture force  out  of  nothing,  any  more  than  we  can  manufacture 
time,  or  space,  or  matter ;  and  in  the  various  machines  for  the 
production  of  power— such  as  the  steam-engine,  the  wind  or 
water  mill,  or  the  electro-motive  machine — we  merely  develop 
or  liberate  the  power  pent  up  in  the  material  which  we  consume 
to  generate  the  power ;  just  as  in  setting  a  clock  in  motion,  we 
liberate  the  power  pent  up  in  the  spring.  Coal  is  virtually  a 
spring  that  has  been  wound  up  by  the  hand  of  nature ;  and  in 
using  it  in  an  engine  we  are  only  permitting  it  to  uncoil — im- 


LAW   OF  VIRTUAL   VELOCITIES.  79 

parting  thereby  to  some  other  agent  an  amount  of  power  equal 
to  that  which  the  coal  itself  loses.  The  natural  agent  employed 
in  winding  up  the  springs  which  our  artificial  machines  uncoil 
is  the  sun,  which  by  its  action  on  vegetation  decomposes  the 
carbonic  acid  which  combustion  produces,  and  uses  the  carbon 
to  build  up  again  the  structure  of  trees  and  plants,  that,  by  their 
subsequent  combustion,  will  generate  power ;  and  as  coal  is  only 
the  fossil  vegetation  of  an  early  epoch,  we  are  now  using  in  our 
engines  the  power  which  the  sun  gave  out  ages  ago.  So  in 
windmills  and  waterwheels,  it  is  the  sun  that,  by  rarefying  some 
parts  of  the  atmosphere  more  than  others,  causes  the  wind  to 
blow  that  impels  windmills,  and  the  vapours  to  exhale,  which, 
being  afterwards  precipitated  as  rain,  form  the  rivers  that  impel 
waterwheels.  In  performing  these  operations  the  sun  must 
lose  as  much  power,  in  the  shape  of  heat  or  otherwise,  as  it  im- 
parts; and  one  of  two  consequences  must  ensue — either  that 
the  sun  is  gradually  burning  out,  or  that  it  is  receiving  back  in 
some  other  shape  the  equivalent  of  the  power  that  it  parts  with. 

LAW  OF  VIRTUAL  VELOCITIES. 

One  branch  of  the  principle  of  conservation  of  force  is  well 
known  in  mechanics  as  the  principle  of  virtual  velocities.  This 
principle  teaches  that,  as  the  power  exerted  in  a  given  time  by  a 
machine,  such  as  a  steam-engine  or  waterwheel,  is  a  definite 
quantity,  and  as  power  is  not  mere  pressure  or  mere  motion,  but 
the  product  of  pressure  and  motion  together,  so  in  any  part  of 
the  machine  that  is  moving  slowly,  the  pressure  will  be  great, 
and  in  any  part  of  the  machine  moving  rapidly,  the  pressure 
must  be  small,  seeing  that  under  no  other  circumstances  could 
the  product  of  the  pressure  and  velocity — which  represents  or 
constitutes  the  power — be  a  constant  quantity.  A  horse  power 
is  a  dynamical  unit,  or  a  unit  of  force,  which  is  represented  by 
33,000  Ibs.  raised  one  foot  high  in  a  minute  of  time;  and  this 
unit  is  usually  called  an  actual  horse  power  to  distinguish  it 
from  the  nominal  or  commercial  horse  power,  which  is  merely 
an  expression  for  the  diameter  of  cylinder  and  length  of  stroke, 


80  MECHANICS    OF   THE    STEAM-ENGINE. 

or  a  measure  of  the  dimensions  of  an  engine  without  any  refer- 
ence to  the  amount  of  power  actually  exerted  by  it.  If  we  sup- 
pose that  an  engine  makes  one  double  stroke  of  5  feet  in  the 
minute — which  is  equal  to  a  space  of  10  feet  in  the  minute  that 
the  piston  must  pass  through,  since  it  has  to  travel  both  upward 
and  downward — and  that  this  engine  when  at  work  exerts  one 
horse  power,  it  is  easy  to  tell  what  pressure  must  be  exerted  on 
the  piston  in  order  that  this  power  may  be  exactly  attained ;  for 
it  must  be  the  10th  of  33,000  or  3,300  Ibs. ;  since  3,300  Ibs.  mul- 
tiplied by  10  feet  is  equivalent  to  33,000  Ibs.  multiplied  by  1 
foot.  Such  an  engine,  if  making  10  strokes  in  the  minute,  would 
exert  10  horses'  power ;  if  making  20  strokes  in  the  minute  would 
exert  20  horses'  power;  if  making  30  strokes  in  the  minute 
would  exert  30  horses'  power ;  and  in  general  the  pressure  on 
the  piston  in  Ibs.  multipled  by  the  space  passed  through  by  the 
piston  in  feet  per  minute,  and  divided  by  33,000,  will  give  the 
number  of  horses1  power  exerted  by  the  engine. 

It  will  be  clear  from  these  considerations  that  the  circum- 
stance which  determines  the  power  exerted  by  any  engine  during 
each  stroke  is — with  any  uniform  pressure  of  steam — the  capacity 
of  the  cylinder.  A  tall  and  narrow  cylinder  will  generate  as 
much  power  each  stroke,  and  will  consume  as  much  steam,  as  a 
short  and  broad  one,  if  the  capacities  of  the  two  are  the  same. 
But  the  strain  to  which  the  piston-rod,  the  working-beam,  and 
the  other  parts  are  subjected,  will  be  greatest  in  the  case  of  the 
short  cylinder,  since  the  weight  or  pressure  on  the  piston  must 
be  greatest  in  that  case  in  order  to  develop  the  same  amount  of 
power.  Since,  too,  in  the  case  of  an  engine  exerting  a  given 
power,  the  quantity  of  power  is  a  constant  quantity,  which  may 
be  represented  by  a  small  pressure  acting  through  a  great  space, 
or  a  great  pressure  acting  through  a  small  space,  so  long  as  the 
product  of  the  space  and  pressure  remain  invariable,  it  follows 
that  in  any  part  of  an  engine  through  which  the  strain  is  trans- 
mitted, and  of  which  the  motion  is  very  slow,  the  pressure  and 
strength  must  be  great  in  the  proportion  of  the  slowness,  since 
the  pressure  multiplied  by  the  motion,  at  any  other  part  of  the 
engine,  must  always  be  equal  to  the  pressure  multiplied  by  the 


LAW   OF  VIRTUAL   VELOCITIES.  81 

motion  of  the  piston.  In  the  case  of  any  part  of  an  engine, 
therefore,  or  in  the  case  of  any  part  of  any  machine  whatever,  it 
is  easy  to  tell  what  the  strain  exerted  will  be  when  we  know  the 
relative  motions  of  the  piston,  or  other  source  of  power,  and  of 
the  part  the  strain  on  which  we  wish  to  ascertain,  since,  if  the 
motion  of  such  part  be  only  i  of  that  of  the  moving  force,  the 
strain  will  be  twice  greater  upon  that  part  than  upon  the  part 
where  the  force  is  first  applied.  If  the  motion  of  the  part  be  -J 
of  that  of  the  moving  force,  the  strain  upon  it  will  be  3  times 
greater  than  that  due  to  the  direct  application  of  the  moving 
force ;  if  the  motion  be  J,  the  strain  will  be  4  times  greater ;  if  j, 
it  will  be  5  times  greater ;  if  TV,  it  will  be  10  times  greater ;  if 
f £o,  it  will  be  100  times  greater :  and  if  any  motion  of  the  prime 
mover  imparts  no  appreciable  motion  to  some  other  part  of  the 
machine,  the  strain  becomes  infinite,  or  would  become  so  only 
for  the  yielding  and  springing  of  the  parts  of  the  machine.  We 
have  an  example  of  a  strain  of  this  kind  in  the  Stanhope  printing 
press,  or  in  the  elbow-jointed  lever,  which  consists  of  two  bars 
jointed  to  one  another  like  the  halves  of  a  two-foot  rule.  If  we 
suppose  these  two  portions  to  be  opened  until  they  are  nearly 
but  not  quite  in  the  same  straight  line,  and  if  they  are  then  in- 
terposed between  two  planes,  and  are  forced  sideways  so  as  to 
bring  them  into  the  same  straight  line,  the  force  with  which  the 
planes  will  be  pressed  apart  will  be  proportional  to  the  relative 
motions  of  the  hand  which  presses  the  elbow-joint  straight,  and 
the  distance  through  which  the  planes  are  thereby  separated. 
As  it  will  be  found  that  this  distance  is  very  small  indeed,  rela- 
tively with  the  motion  of  the  hand,  when  the  two  portions  of 
the  lever  come  nearly  into  the  same  straight  line,  and  ceases 
altogether  when  they  are  in  the  same  straight  line,  so  the  pressure 
acting  in  separating  the  planes  will  be  very  great  indeed  when 
the  parts  of  the  lever  come  into  nearly  a  straight  line,  and  is  in- 
finite when  they  come  really  into  a  straight  line ;  or  it  would  be 
so  but  for  the  compressibility  of  the  metal  and  the  yielding  of  the 
parts  of  the  apparatus. 

It  is  perfectly  easy,  with  the  aid  of  the  law  of  virtual  veloci- 
ties, to  determine  the  strains  existing  at  any  part  of  a  machine, 
4* 


82  MECHANICS   OF   THE   STEAM-ENGINE. 

and  also  the  weight  which  the  exertion  of  any  given  force  at  the 
handle  of  a  crane,  winch,  screw,  hydraulic  press,  differential 
screw,  blocks  and  tackle,  or  any  other  machine  will  lift ;  for  wo 
have  only  to  determine  the  first  and  last  velocities,  and  in  the 
proportion  in  which  the  last  velocity  is  slow,  the  weight  lifted 
will  be  great.  Thus,  suppose  we  have  a  crane,  moved  by  a  han- 
dle which  has  a  radius  of  2  feet,  which  turns  a  pinion  of  6  inches 
diameter  gearing  into  a  wheel  of  4  feet  diameter,  on  which  there 
is  a  barrel  of  1  foot  diameter  for  winding  the  chain  upon,  it  is 
easy  to  tell  what  weight — excluding  friction — will  be  balanced 
or  lifted  by,  say  a  force  of  30  Ibs.  applied  at  the  handle.  The 
handle,  it  is  clear,  will  describe  a  circle  of  4  feet  diameter,  while 
the  pinion  describes  only  a  circle  of  6  inches  diameter,  which 
gives  us  a  relative  velocity  of  8  to  1 ;  or,  in  other  words,  the 
strain  exerted  at  the  circumference  of  the  pinion  will  be  8  times 
greater  than  the  strain  of  30  Ibs.  applied  at  the  end  of  the  handle ; 
so  that  it  will  be  240  Ibs.  Now  the  strain  of  the  pinion  is  im- 
parted to  the  circumference  of  the  wheel  with  which  it  gears ; 
and  the  strain  of  240  Ibs.  at  the  circumference  of  a  wheel  of  4 
feet  diameter  will  be  4  times  greater  at  the  circumference  of  a 
barrel  of  1  foot  diameter,  placed  on  the  same  shaft  as  the  wheel, 
and  revolving  with  it.  The  weight  on  the  barrel,  therefore, 
which  will  balance  30  Ibs.  on  the  handle,  will  be  4  times  240  Ibs., 
or  960  Ibs.,  but  for  every  foot  through  which  the  weight  of 
960  Ibs.  is  raised,  the  handle  must  move  through  32  feet,  since 
30  Ibs.  moved  through  32  feet  is  equivalent  to  960  Ibs.  moved 
through  1  foot.  So  also  in  the  case  of  a  screw  press,  the  screw 
of  which  has  a  pitch  of  say  half  an  inch,  and  which  is  turned 
round  by  a  lever  say  3  feet  long,  pressed  with  a  weight  of  30  Ibs. 
on  the  end  of  it,  we  have  here  a  moving  force  acting  in  a  circle 
of  6  feet  diameter ;  and  as  at  each  revolution  of  the  screw  it  is 
moved  downward  through  a  distance  equal  to  the  pitch,  which 
is  %  inch,  we  have  the  relative  velocities  of  •£  inch,  and  the  cir- 
cumference of  a  circle  6  feet  in  diameter.  Now  the  proportion 
of  the  diameter  of  a  circle  to  its  circumference  being  1  to 
3-1416,  the  circumference  of  a  circle  6  feet  diameter  will  be 
18-8496  feet,  or  say  18-85  feet,  which,  multipled  by  12  to  reduce 


MODE   OF   COMPUTING   STRAINS.  83 

it  to  inches,  since  the  pitch  is  expressed  in  inches,  gives  us  226'2 
inches,  and  the  relative  velocities,  therefore,  are  22 6 '2  to  -J-,  or 
452-4  to  1.  It  follows,  consequently,  that  a  pressure  of  30  Ihs. 
applied  at  the  end  of  the  lever  employed  to  turn  such  a  screw  as 
has  been  here  supposed,  will  produce  at  the  point  of  the  screw  a 
pressure  of  452'4  times  30,  or  13,572  Ibs.,  which  is  a  little  over  6 
tons.  Whatever  the  species  of  mechanism  may  he — whether  a 
hydraulic  press,  a  lever,  ropes  and  pulleys,  differential  wheels, 
screws,  or  pulleys,  or  any  other  machine  or  apparatus,  this  in- 
variable law  holds,  that  with  any  given  pressure  or  strain  at  the 
point  where  the  motion  begins,  the  pressure  or  strain  exerted  at 
any  part  of  the  machine  will  be  in  the  inverse  proportion  of  its 
velocity — the  stress  or  pressure  on  any  part  being  great,  just  in 
the  proportion  hi  which  its  motion  is  slow. 

In  the  case  of  a  lever  like  the  beam  of  a  pair  of  scales,  which 
has  its  fulcrum  in  the  middle  of  its  length,  the  application  of 
any  force  or  pressure  at  one  end  of  the  beam  will  produce  an 
equal  force  or  pressure  at  the  other  end ;  and  both  of  the  ends 
will  also  move  through  the  same  distance  if  motion  be  given  to 
either.  But  if  the  fulcrum,  instead  of  being  placed  in  the  mid- 
dle of  the  beam,  be  placed  intermediately  between  the  middle 
and  one  end,  we  shall  then  have  a  lever  of  which  the  long  end 
is  3  times  the  length  of  the  short  one,  and  a  pound  weight 
placed  at  the  extremity  of  the  long  end,  will  balance  3  Ibs. 
weight  placed  at  the  extremity  of  the  short  end.  If,  however, 
the  short  end  be  moved  through  1  foot,  the  long  end  will  be 
simultaneously  moved  through  3  feet;  and  3  Ibs.  gravitating 
through  1  foot  expresses  just  the  same  amount  of  mechanical 
power  as  1  Ib.  gravitating  through  3  feet.  In  a  safety-valve, 
pressed  down  by  a  lever  5  feet  long,  while  the  point  which 
presses  on  the  spindle  of  the  safety-valve  is  6  inches  distant 
from  the  fulcrum,  we  have  a  lever,  the  ends  of  which  have  a 
proportion  of  i  to  5,  or  1  to  10 ;  so  that  every  pound  weight 
hung  at  the  extremity  of  the  long  end  of  such  a  lever,  will  be 
equivalent  to  a  weight  of  10  Ibs.  placed  on  the  top  of  the  valve 
itself.  In  the  case  of  a  set  of  blocks  and  tackle,  say  with  3 
sheaves  in  each  block,  and,  therefore,  with  6  ropes  passing  from 


84  MECHANICS    OF   THE    STEAM-ENGINE. 

one  block  to  the  other,  it  is  clear  that  if  the  weight  to  be  lifted 
be  raised  a  foot,  each  of  the  ropes  will  have  been  shortened  a 
foot,  to  do  which — as  there  are  6  ropes — the  rope  to  which  the 
motive  power  is  applied  must  have  been  pulled  out  6  feet.  We 
have,  here,  therefore,  a  proportion  of  6  to  1 ;  or,  in  other  words, 
a  weight  of  1  cwt.  applied  to  the  rope  which  is  pulled,  would 
balance  6  cwt.  suspended  from  the  blocks. 

It  is  a  common  practice  among  sailors  in  tightening  ropes — 
after  having  first  drawn  the  rope  as  far  as  they  can  by  pulling  it 
towards  them — to  pass  the  end  of  the  rope  over  some  pin  or 
other  object,  and  then  to  pull  it  sideways  in  the  manner  a  harp 
string  is  pulled,  taking  in  the  slack  as  they  again  release  it. 
This  action  is  that  of  the  elbow-jointed  lever  reversed;  and  inas- 
much as  the  tightened  rope  may  be  pulled  to  a  considerable  dis- 
tance sideways,  without  any  appreciable  change  in  its  total 
length,  the  strain  imparted  by  this  side  pulling  is  great  in  the 
proportion  of  the  smallness  of  the  distance  through  which  any 
given  amount  of  side  deflection  will  draw  the  rope  on  end. 

A  hydraulic  press  is  a  machine  consisting  of  a  cylinder  fitted 
with  a  piston,  beneath  which  piston  water  is  forced  by  a  small 
pump ;  and  at  each  stroke  of  the  pump  the  piston  or  ram  of  the 
hydraulic  cylinder  is  raised  through  a  small  space,  which  will 
be  equal  to  the  capacity  of  the  pump  spread  over  the  area  of 
the  hydraulic  piston.  If,  for  example,  the  pump  has  an  area  of 
1  square  inch,  and  a  stroke  of  12  inches,  its  capacity  or  content 
will  be  12  cubic  inches ;  and  if  the  piston  has  an  area  of  144 
square  inches,  it  is  clear  that  the  pump  must  empty  itself  12 
times  to  project  144  cubic  inches  of  water  into  the  cylinder,  and 
which  would  raise  the  piston  or  ram  1  inch.  In  other  words, 
the  plunger  of  the  pump  must  pass  through  12  times  12  inches, 
or  144  inches,  to  raise  the  piston  of  the  hydraulic  cylinder  1 
inch,  so  that  the  motion  of  the  piston  or  ram  of  the  hydraulic 
cylinder  being  144  tunes  slower  than  that  of  the  plunger  of  the 
pump,  it  will  exert  144  times  the  pressure  that  is  exerted  on  the 
piston  of  the  pump  to  move  it.  When,  therefore,  we  know  the 
amount  of  pressure  that  is  applied  to  move  the  plunger  of  the 
pump,  we  can  easily  tell  the  weight  that  the  hydraulic  piston 


MODE    OF    COMPUTING    STRAINS.  85 

will  lift,  or  the  pressure  that  it  will  exert ;  and,  indeed,  this 
pressure  will  be  greater  than  that  on  the  pump  in  the  proportion 
of  the  greater  area  of  the  hydraulic  piston,  relatively  with  that 
of  the  pump  plunger,  and  which  in  the  case  supposed  is  144  to  1. 

There  are  various  forms  of  differential  apparatus  for  raising 
weights,  or  imparting  pressure,  in  which  the  terminal  motion  is 
rendered  very  slow,  and  therefore  the  terminal  pressure  very 
great,  by  providing  that  it  shall  be  the  difference  of  two  mo- 
tions, very  nearly  equal,  but  acting  in  opposite  directions. 
Thus,  if  the  bight  of  a  rope  be  made  to  hang  between  two 
drums  or  barrels  on  which  the  different  ends  of  the  rope  are 
wound,  and  one  of  which  barrels  pays  the  rope  out,  while  the 
other  winds  it  up  at  a  slightly  greater  velocity  than  that  with 
which  it  is  unwound  by  the  other,  the  bight  of  the  rope  will  be 
very  slowly  tightened ;  and  any  weight  hung  upon  the  bight 
will  be  lifted  up  with  a  correspondingly  great  force.  Then  there 
are  forms  of  the  screw  press  in  which  the  screw  winds  itself  up 
a  certain  distance  at  one  end,  and  unwinds  itself  nearly  the 
same  distance  at  the  other  end ;  so  that,  at  each  revolution,  it 
advances  the  object  it  presses  upon  through  a  distance  equal  to 
the  difference  of  the  winding  and  unwinding  pitches ;  and  as 
this  difference  may  be  made  as  small  as  we  please,  so  the  pres- 
sure may  be  made  as  great  as  we  please.  The  effect  of  using 
these  differential  screws  is  the  same  as  would  be  obtained  if  we 
were  to  use  a  single  common  screw  having  a  pitch  equal  to  the 
differences  of  the  pitches.  But  in  practice  such  a  pitch  would 
be  too  fine  to  have  the  necessary  strength  to  resist  the  pressure ; 
and  consequently  differential  screws  are  in  every  respect  prefer- 
able. 

It  is  easy  to  tell  what  the  pressure  exerted  by  a  differential 
screw  will  be,  when  we  know  the  actual  advance  it  makes  at 
each  revolution.  Thus,  suppose  the  pitch  of  the  unwinding  or 
screwing-out  part  of  the  screw  to  be  half  an  inch,  or  -ffifa  of  an 
inch,  and  the  pitch  of  the  winding  or  screwing-in  part  of  the 
screw  to  be  -f^y  of  an  inch,  then  the  distance  between  the 
winding  and  unwinding  nuts  will  be  increased  -nftfr— -j^jftfr  or 
inch  at  each  revolution.  The  pressure  exerted 


86  MECHANICS   OF   THE   STEAM-ENGINE. 

by  such  a  screw  will  consequently  be  the  same  as  if  the  pitch 
were  TnVvth  Part  °f  an  mch  ;  and  such  pressure  may  be  easily 
computed  in  the  manner  already  explained. 

There  are  various  forms  of  differential  gearing  employed  in 
special  cases — not  generally  for  the  purpose  of  generating  a 
great  pressure,  but  for  the  purpose  of  generating  a  slow  motion 
with  few  wheels ;  though  a  great  pressure  is  an  incident  of  the 
arrangement,  if  the  terminal  motion  be  resisted.  Thus,  if  we 
place  two  bevel  wheels  on  the  same  shaft,  with  the  teeth  facing 
one  another,  and  cause  the  two  wheels  to  make  the  same  num- 
ber of  revolutions  in  opposite  directions,  and,  further,  if  we 
place  between  the  two  wheels,  and  on  the  end  of  a  crank  or 
arm  capable  of  revolving  between  them,  a  bevel  pinion,  gearing 
with  the  two  wheels,  then  it  will  follow — if  the  two  wheels 
have  the  same  number  of  teeth — that  the  bevel  pinion  will 
merely  revolve  on  its  axis,  but  that  this  axis  or  crank  will  be 
itself  stationary.  If,  however,  one  wheel  is  made  with  a  tooth 
more  than  the  other  wheel,  then  it  will  follow  that  the  crank  or 
arm  carrying  the  bevel  pinion  will  be  advanced  through  the  dis- 
tance of  one  tooth  by  each  revolution  of  the  wheels,  and  the 
arm  will  consequently  have  a  very  slow  motion  round  the  shaft, 
and  will  impart  a  correspondingly  great  pressure  to  any  object 
by  which  that  motion  is  resisted.  Differential  gearing  is  princi- 
pally employed  for  drawing  along,  very  slowly,  the  cutter  block 
in  boring  mills ;  and  many  of  its  forms  are  very  elegant.  It  is 
also  employed  in  various  kinds  of  apparatus  for  recording  the 
number  of  strokes  made  by  an  engine  in  a  given  time.  But  the 
same  conditions  which  render  the  motion  slow,  also  render  it 
forcible ;  without  any  reference  to  the  forms  of  apparatus  by 
which  the  transformation  is  produced. 

These  expositions  are  probably  sufficient  to  show  how  the 
pressure  exerted  by  any  machine  may  be  computed ;  and  as  the 
pressure  is  only  another  name  for  the  strain,  we  may  thence 
discover  how  to  apportion  the  material  to  give  the  necessary 
strength.  The  very  same  considerations  will  enable  us  to  deter- 
mine the  strains  existing  at  any  part  of  an  engine,  or  at  any 
part  of  any  structure  whatever ;  and  when  we  know  the 


MODE    OF    COMPUTING    STRAINS.  87 

amount  of  the  strain,  it  becomes  easy  to  tell  how  much  mate- 
rial, of  any  determinate  strength,  we  must  apply  in  order  to  re- 
sist it.  Let  us  suppose,  for  example,  that  we  wished  to  know 
the  strain  which  exists  at  any  part  of  the  main  heam  of  a  land 
engine,  in  order  that  we  may  determine  what  quantity  of  metal 
we  should  introduce  into  it  to  give  it  the  necessary  strength. 
Now  if  we  suppose  the  fly  wheel  to  be  jammed  fast  when  the 
steam  is  put  on  the  engine,  it  is  clear  that  the  connecting-rod 
end  of  the  beam  will  be  thereby  fixed,  and  will  become  a  ful- 
crum round  which  the  piston-rod  will  endeavour  to  force  up  the 
beam,  lifting  the  main  centre  with  twice  the  pressure  that  the 
piston  exerts ;  since  if  we  suppose  the  main  centre  to  be  a 
weight,  and  the  fulcrum  to  be  at  the  end  of  the  beam,  this 
weight  would  only  be  moved  through  one  inch,  when  the  piston 
moved  through  2  inches,  so  that  the  lifting  pressure  upon  this 
point  would  be  twice  greater  than  that  upon  the  piston,  and  the 
main  centre  must  consequently  be  made  strong  enough  to  with- 
stand this  strain.  If,  however,  we  suppose  the  main  centre  to 
be  sufficiently  strong,  we  may  dismiss  all  consideration  respect- 
ing it,  and  may  consider  the  beam,  which  will  be  thus  fixed  at 
two  points,  as  a  beam  projecting  from  a  wall,  which  an  upward 
or  downward  pressure  is  applied  to  break. 

Now  in  any  well-formed  engine  beam,  and  indeed  in  all 
metal  beams  of  proper  construction,  the  strength  is  collected  at 
the  edges ;  and  the  web  of  the  beam  acts  merely  in  binding 
into  one  composite  mass  the  areas  of  metal  which  are  to  be 
compressed  and  extended.  The  edges  of  the  beam  may  be  in 
fact  regarded  as  pillars,  which  it  is  the  tendency  of  the  strain 
applied  to  the  beam  to  crumple  up  on  the  one  edge,  and  tear 
asunder  on  the  other  edge ;  and  the  whole  strength  of  the  beam 
may  be  supposed  to  reside  in  these  pillars,  since  if  they  were  to 
break  the  rest  of  the  beam  would  at  once  give  way.  The 
strength  of  any  given  material  to  resist  compression  is  not  neces- 
sarily, nor  always  the  same  as  the  strength  to  resist  compression. 
In  the  case  of  wrought-iron  the  stretching  strength  is  about 
twice  greater  than  the  crumpling  strength  ;  whereas,  in  the  case 
of  cast-iron  the  crushing  strength  is  between  5  and  6  tunes 


00  MECHANICS    OF    THE    STEAM-ENGINE. 

greater  than  the  tensile  strength.  In  the  case  of  an  engine 
beam,  which  has  the  strain  applied  alternately  in  each  direction, 
the  weakest  strength  must  necessarily  be  that  on  which  our 
computations  are  based ;  and  in  machinery  it  is  not  advisable  to 
load  cast-iron  with  a  greater  weight  than  2,000  Ibs.  per  square 
inch  of  section.  Now  if  we  suppose,  for  the  sake  of  simplify- 
ing the  computation,  that  the  depth  of  the  beam  at  the  centre 
is  equal  to  its  length,  then  it  is  clear  that  if  the  end  of  the  beam 
moves  through  any  given  distance,  a  point  on  the  edge  of  the 
beam  over  or  below  the  main  centre  will  move  through  the 
same  distance,  having  the  same  radius ;  and  if  we  suppose  that 
the  depth  of  the  beam  is  equal  to  half  its  length,  then  a  point 
on  the  edge  of  the  beam,  over  or  below  the  main  centre,  will 
move  through  half  the  space  that  the  end  of  the  beam  moves 
through,  and  at  such  point  there  will  consequently  be  twice  the 
amount  of  strain  existing  than  is  exerted  upon  the  piston.  For 
every  2,000  Ibs.,  therefore,  of  pressure  on  the  piston,  there 
ought  to  be  strength  enough  at  the  edge  of  the  beam  to  with- 
stand a  strain  of  4,000  Ibs. ;  but  as  this  strength  has  to  be  di- 
vided between  the  two  edges  of  the  beam,  there  should  be 
strength  enough  at  each  end  to  bear  2,000  Ibs.  without  straining 
the  metal  more  than  2,000  Ibs.  per  square  inch  of  section.  In 
other  words,  with  such  a  proportion  of  beam  there  ought  to  bo 
a  square  inch  of  section  in  the  top  and  bottom  flanges  or  mould- 
ings of  the  beam,  for  each  2,000  Ibs.  pressure  or  load  upon  the 
piston.  In  land  engines  a  common  proportion  for  the  depth  of 
the  beam  is  the  diameter  of  the  cylinder ;  and  a  common  pro- 
portion for  the  length  of  stroke  is  twice  the  diameter  of  the  cyl- 
inder, while  the  length  of  the  beam  is  commonly  made  equal  to 
three  times  the  length  of  the  stroke.  With  these  proportions 
the  length  of  the  beam  will  be  equal  to  six  times  its  depth ;  and 
as  the  edge  of  the  beam,  above  or  below  the  main  centre,  will 
in  such  a  beam  have  only  one-sixth  of  the  motion  that  the  end 
of  the  beam  has,  the  strain  at  that  part  divided  between  the 
two  edges  of  the  beam  will  be  six  times  as  great  as  the  stress 
exerted  on  the  piston.  For  every  2,000  Ibs.  pressure,  therefore, 
on  the  piston,  there  must  be  about  three  square  inches  of  sec- 


CASES    IN    WHICH    STRAINS    ARE    INFINITE.  89 

tional  area  in  the  upper  and  lower  flanges  or  mouldings  of  the 
beam,  or  six  square  inches  between  the  two ;  while  the  web  of 
the  beam  is  made  merely  strong  enough  to  keep  the  upper  and 
lower  flanges  in  their  proper  relative  positions. 

It  will  be  obvious  from  these  considerations,  that  the  prin- 
ciple of  virtual  wlocities  enables  us  to  compute  the  amount  of 
strain  existing  at  any  part  of  any  machine  or  engine,  as  we  have 
only  to  suppose  the  part  to  be  broken,  and  to  see  what  amount 
of  motion  the  broken  part  will  have  relatively  with  the  motion 
of  the  prime  mover,  to  determine  the  amount  of  the  strain.  We 
can  also  easily  discern,  by  keeping  this  principle  in  view,  how  it 
comes  that,  in  the  case  of  marine  or  other  engines  arranged  in 
pairs,  with  the  cranks  at  right  angles  with  one  another,  one  of 
the  engines  is  so  often  broken  by  water  getting  into  the  cylin- 
der ;  and  how  necessary,  therefore,  it  is  that  such  engines  should 
be  provided  with  safety-valves,  so  enable  the  water  shut  within 
the  cylinder  to  escape.  For  if  water  gets  into  one  cylinder,  and 
if  at  or  near  the  end  of  the  stroke  the  slide-valve  shuts  off  the 
communication  both  with  the  boiler  and  with  the  condenser,  as 
is  a  common  state  of  things,  it  will  follow  that  the  water  shut 
within  the  cylinder,  being  unable  to  escape,  will  resist  the  de- 
scent of  the  piston.  As,  moreover,  the  crank  of  one  engine  is 
vertical,  while  that  of  the  other  is  horizontal,  and  as  when  ver- 
tical the  crank  is  virtually  an  elbow-jointed  lever,  it  will  follow 
that  one  engine,  with  its  greatest  leverage  of  crank,  is  moving 
into  the  vertical  position  the  crank  of  the  other  engine,  in  which 
position  it  will  act  like  an  elbow-jointed  lever,  or  the  lever  of  a 
Stanhope  press,  in  forcing  down  the  piston  on  the  water,  with  a 
pressure  that  is  infinite ;  and  as  the  water  is  nearly  incompress- 
ible, and  as  in  the  absence  of  escape- valves  it  cannot  get  away, 
some  part  of  the  engine  must  necessarily  break.  The  smaller 
the  quantity  of  water  shut  within  the  cylinder,  so  long  as  it  re- 
sists the  piston,  the  greater  the  breaking  pressure  will  be ;  as  the 
crank  will,  in  such  case,  come  more  nearly  into  the  vertical  po- 
sition where  the  downward  thrust  that  it  exerts  is  greatest; 
whereas,  if  there  be  any  large  volume  of  water  shut  within  the 
cylinder,  the  piston  will  encounter  it  before  the  crank  cornea 


90  MECHANICS   OF  THE    STEAM-ENGINE. 

near  the  vertical  position,  and  also  before  the  crank  of  the  other 
engine  comes  into  the  horizontal  position  in  which  it  exerts  the 
greatest  leverage  in  turning  round  the  shaft,  as  it  does  when  the 
engine  is  at  half  stroke.  In  these  as  in  all  other  cases  in  which 
we  wish  to  investigate  the  strain  produced  in  any  machine,  or  in 
any  part  of  any  machine,  by  any  given  pressure  applied  in  any 
direction,  whether  oblique  or  otherwise,  we  have  only  to  con- 
sider the  amount  of  motion — in  the  direction  in  which  the  strain 
acts — of  that  particular  part  which  endures  the  strain  or  com- 
municates the  pressure,  relatively  with  the  amount  of  simulta- 
neous motion  in  the  prime  mover.  And  if  the  ultimate  motion 
be  a  tenth,  a  hundredth,  or  a  thousandth  part  of  the  original 
motion,  so  will  the  strain  or  pressure  exerted  by  the  prime  mover 
at  the  part  where  the  motion  is  first  communicated  be  multiplied 
ten,  a  hundred,  or  a  thousand  fold. 

NATURE  OF  MECHANICAL  POWER. 

Mechanical  power,  or,  as  it  is  sometimes  defined,  work,  or  vis 
viva,  is  pressure  acting  through  space ;  and  the  law  of  the  con- 
servation of  force  teaches  that  power  once  produced  cannot  be 
annihilated,  though  it  may  be  transformed  into  other  forces  of 
equivalent  value.  In  all  machines  a  certain  proportion  of  the 
power  resident  in  the  prime  mover  is  lost,  while  the  rest  is  util- 
ised and  is  rendered  available  for  the  performance  of  those 
labours  for  which  power  is  required.  Thus,  in  a  waterwheel,  the 
theoretical  value  of  the  fall  is  that  due  to  a  certain  weight  of 
water  gravitating  through  a  certain  number  of  feet  in  the  min- 
ute ;  and  if  we  know  the  height  of  the  fall,  and  the  discharge 
of  water  in  a  given  time,  the  theoretical  value  of  such  a  fall  can 
be  easily  computed.  But  by  no  species  of  hydraulic  instrument, 
whether  a  waterwheel,  a  turbine,  a  water-pressure  engine,  a 
Barker's  mill,  or  any  other  machine,  can  the  whole  of  the  power 
be  abstracted  from  the  fall,  and  be  made  available  for  useful 
purposes.  About  80  per  cent,  of  the  theoretical  power  of  a 
waterfall  is  considered  to  be  a  very  satisfactory  result  to  obtain 
in  practice ;  and  the  rest  is  lost  by  impact  and  eddies,  and  by 


MECHANICAL  EQUIVALENT  OF  HEAT.         91 

the  friction  of  the  water  and  of  the  machine.  In  the  steam- 
engine  the  motive  force  is  not  gravity,  but  heat ;  and  just  in  the 
same  way  as  power  is  imparted  by  water  in  descending  from  a 
higher  to  a  lower  level,  so  is  power  imparted  by  heat  in  descend- 
ing from  a  higher  to  a  lower  temperature.  These  two  tempera- 
tures are  the  the  temperature  of  the  boiler,  and  the  temperature 
of  the  condenser ;  and  it  is  clear  that  if  the  condenser  were  to 
be  made  as  hot  as  the  boiler,  the  motion  of  the  engine  would 
cease.  And  just  as  in  a  waterfall  there  is  a  certain  theoretical 
power  due  to  the  quantity  of  gravitating  matter  and  the  differ- 
ence of  level,  so  in  a  steam-engine  there  is  also  a  certain  theoret- 
ical power  due  to  the  quantity  of  heated  matter,  and  the  differ- 
ence of  temperature ;  but  in  utilising  the  power  of  steam-en- 
gines, this  theoretical  limit  is  not  approached  so  nearly  as  in  hy- 
draulic machines.  The  great  fault  of  the  steam-engine  is  that 
the  larger  part  of  the  attainable  fall  is  lost.  Thus,  if  we  sup- 
pose the  temperature  of  the  furnace  to  be  2,500°  Fahrenheit,  and 
the  temperature  of  the  boiler  to  be  250°,  while  that  of  the  con- 
denser is  100°,  we  utilise  pretty  effectually  the  power  represented 
by  the  difference  in  temperature  between  100°  and  250° ;  but 
the  difference  between  250°  and  2,500°  is  not  utilised  at  all. 
The  consequence  of  this  state  of  things  is  that  not  above  one- 
tenth  of  the  power  theoretically  due  to  the  fuel  consumed,  is 
utilised  in  the  best  modern  steam-engines — the  rest  being 
thrown  away. 

MECHANICAL   EQUIVALENT  OF  HEAT. 

If  the  law  of  the  conservation  of  force  be  an  invariable  law 
of  nature,  we  shall  naturally  expect  to  find  that  the  power  which 
is  consumed  when  a  steam-engine  or  other  machine  is  set  to  ex- 
ecute useful  work,  reappears  as  an  equivalent  force  in  some 
other  form.  This  consequently  is  the  case.  When  an  engine  is 
employed  to  pump  water,  we  have  obviously  the  equivalent  of 
the  force  in  the  water  pumped  to  a  higher  level ;  and  if  this 
water  were  suffered  to  flow  back  again,  so  as  in  its  descent  to 
generate  power,  we  should  again  have  the  power  we  before 


92  MECHANICS    OF   THE    STEAM-ENGINE. 

spent,  with  the  deductions  due  to  the  imperfections  of  the  appa- 
ratus employed.  In  the  case  of  an  engine,  however,  which  ex- 
pends its  power  in  friction,  or  in  such  work  as  the  propulsion 
of  a  vessel  through  the  water,  the  reproduction  of  the  equivalent 
of  the  power  expended  is  not  so  easily  perceived.  But  in  these 
cases,  also,  it  has  been  proved  by  careful  experiment,  that  the 
law  of  the  conservation  of  force  equally  obtains.  Friction, 
whether  of  solids  or  liquids,  produces  heat ;  and  in  the  case  of 
an  engine  which  expends  its  power  on  a  friction  brake,  or  on 
any  other  analogous  object,  an  amount  of  heat  will  be  produced, 
such  as,  if  it  could  be  used  without  loss  in  a  perfect  engine, 
would  exactly  reproduce  the  amount  of  power  expended.  In 
the  case  of  a  vessel  propelled  through  the  water,  the  power  is 
mainly  consumed  in  overcoming  the  friction  of  the  water  on  the 
bottom  of  the  vessel,  and  a  part  is  also  expended  in  moving  the 
water  to  a  greater  or  less  extent ;  and  whatever  motion  the 
water  acquires,  implies  a  corresponding  loss  of  power  by  the  en- 
gine, which  power  is  ultimately  expended  in  moving  the  parti- 
cles of  water  upon  one  another.  In  such  operation  heat  is  pro- 
duced; which  heat,  if  it  could  be  utilised  without  loss  in  an 
engine,  would  exactly  reproduce  the  power  expended.  It  has 
been  found  by  careful  experiment,  that  if  the  power  developed 
by  the  descent  of  a  pound  weight  through  772  feet  be  expended 
in  agitating  a  pound  of  water,  it  will  raise  the  temperature  of 
that  water  1°  Fahrenheit.  The  fall  of  any  given  quantity  of 
water  through  772  feet  is  consequently  called  the  Mechanical 
Equivalent  of  the  heat  required  to  raise  the  same  quantity  of 
water  one  degree  in  temperature ;  since  theoretically  the  two 
values  are  equivalent,  and  practically  the  power  will  produce 
the  heat.  But  we  have  not  yet  any  form  of  apparatus  by  which 
the  heat  would  produce  the  power ;  and  before  we  can  possess 
such,  we  must  have  an  engine  ten  times  better  than  the  best 
form  of  steam-engine  at  present  in  use.  There  is  every  reason 
to  believe  that  there  is  a  definite  quantity  of  mechanical  power 
or  energy  in  the  universe,  the  amount  of  which  can  neither  be 
increased  nor  diminished,  though  it  may  be  transformed  from 
one  shape  into  another ;  and  heat,  light,  electricity,  and  all 


LAWS    OF   FALLING   BODIES.  93 

chemical  and  vital  phenomena  are  merely  phases,  more  or  less 
complex  and  disguised,  of  the  same  elementary  force. 

LAWS  OF  FALLIXG  BODIES. 

Bodies  falling  to  the  earth  by  gravity  are  drawn  thither  by  a 
species  of  attraction — constant  in  amount — which  acts  in  a  man- 
ner similar  to  that  which  reveals  itself  when  two  bodies  in  op- 
posite electrical  states  are  brought  into  proximity.  "We  do  not 
know  with  any  certainty  the  cause  of  gravity.  But  we  know 
that  it  would  be  quite  impossible  for  one  body  to  act  upon 
another  without  some  link  to  connect  the  two  together ;  and  the 
most  probable  supposition  is,  that  as  sound  is  a  pulsation  of  the 
air,  caused  by  pulsations  of  the  sounding  body,  and  as  light  is  a 
pulsation  in  the  ether  which  fills  all  space,  caused  by  pulsations 
of  the  illuminating  body,  so  gravity  is  a  similar  pulsation  in  the 
ether,  or  a  pulsation  in  another  kind  of  ether,  caused  by  the  pul- 
sations of  the  attracting  body.  We  know  by  experience  that  sim- 
ilar pulsations  may  be  generated  in  a  piece  of  iron  by  sending  an 
electric  current  through  it  under  certain  conditions,  and  which, 
for  the  time,  transforms  the  iron  into  a  magnet,  which  will  at- 
tract iron  in  the  same  way  in  which  the  earth  attracts  heavy 
bodies :  and,  in  like  manner,  a  piece  of  amber  or  of  sealing-wax 
may  be  made  to  attract  straws,  pieces  of  paper,  and  other  light 
substances,  by  being  briskly  rubbed.  The  phenomena  of  the 
gyroscope  seem  to  show  that  gravity  takes  an  appreciable  time 
to  act.  If  a  heavy  wheel  set  on  the  end  of  a  horizontal  shaft, 
which  is  sustained  by  two  suitable  supports,  be  put  into  rapid 
rotation,  the  support  nearest  the  wheel  may  be  taken  away 
without  the  wheel  falling  down,  from  which  it  appears  that  the 
pulsations  which  produce  gravity  may  be  so  confounded  together 
by  the  rapid  change  in  the  position  of  the  wheel,  and  conse- 
quently in  the  rapid  change  in  the  direction  of  the  attracting 
pulses  or  waves,  that  the  phenomena  of  gravity  are  no  longer 
exhibited,  or  what  remains  of  them  is  manifested  in  a  horizontal 
direction  instead  of  in  a  vertical — the  wheel  having  shifted  into 
or  towards  that  direction  before  the  pulsations  have  had  time 


94  MECHANICS    OF   THE    STEAM-ENGINE. 

to  be  completed.  We  know  from  experience  that  conflicting 
sounds  may  be  made  to  produce  silence,  and  that  conflicting 
lights  may  be  made  to  produce  darkness ;  and  in  like  manner,  it 
would  appear,  that  a  conflict  in  the  pulsations  which  are  the 
cause  of  gravity  may  sensibly  impair  or  destroy  that  gravity. 
It  has  long  been  known  that  sunlight  consists  of  light  of  those 
different  colours  which  are  exhibited  in  the  rainbow,  and  that 
the  phenomena  of  colours  in  natural  objects  is  produced  by  the 
property  those  objects  have  of  absorbing  some  rays,  and  reflect- 
ing others,  so  that  in  a  red  object  the  whole  of  the  rays  except 
the  red  rays  are  absorbed — and  they  are  reflected;  and  in  a 
blue  object  the  whole  of  the  rays  except  the  blue  rays  are  ab- 
sorbed— and  they  are  reflected ;  and,  as  only  the  reflected  rays 
meet  the  eye,  the  objects  appear  of  a  red  or  blue  colour.  It  has 
also  long  been  known  that  in  black  objects  the  whole  of  the  rays 
are  absorbed,  and  none  reflected ;  and  in  white  objects  that  the 
whole  are  reflected  and  none  absorbed.  But  the  resources  of 
photography  also  enable  us  to  know  that  there  is  a  species  of 
light  which  is  invisible — which  has  no  colour,  and  no  illuminat- 
ing power,  but  which  reveals  its  existence  by  the  effect  it  pro- 
duces on  photographic  preparations.  The  use  of  these  photo- 
graphic preparations  is  consequently  equivalent  to  the  acquisition 
of  a  distinct  sense ;  and  one  of  the  most  important  problems  in 
philosophy  is  to  discover  how  we  may  acquire  the  use  of  artificia. 
senses,  whereby  we  may  more  effectually  interrogate  nature. 
There  may  be  rays  in  sunlight,  and  modes  of  communication  be- 
tween one  body  and  another,  of  which  we  have  no  distinct  con- 
ception yet ;  but  there  must  be  a  mode  of  communication,  of 
some  kind  or  other,  in  every  case  in  which  cause  and  effect  are 
known  to  exist. 

The  force  of  gravity,  like  the  force  of  light  or  of  sound,  varies 
in  strength  with  the  extension  of  the  orb  of  propagation ;  or,  in 
other  words,  it  diminishes  in  intensity  according  to  a  given  law 
with  the  distance  from  the  earth's  surface.  Nor  is  this  force 
precisely  the  same  in  all  parts  of  the  world,  as  near  the  equator 
it  is  partly  counteracted  by  the  operation  of  the  centrifugal 
force  due  to  the  earth's  rotation.  But  all  these  disturbing  causes 


LAWS   OF   FALLING   BODIES.  95 

are  of  too  little  effect  to  be  worth  noticing  further  in  a  work  of 
this  kind;  and  for  all  practical  purposes  we  may  reckon  the 
force  of  gravity  as  uniform  in  all  ages,  and  at  all  parts  of  the 
earth's  surface.  Now,  as  power  is  pressure  acting  through 
space,  a  falling  body  just  before  it  reaches  the  earth  must  have 
a  certain  proportion  of  mechanical  power  stored  up  in  it  which, 
if  again  used  to  raise  the  weight,  would  carry  it  up  once  more  to 
its  original  position.  This  action  we  observe  in  a  pendulum.  If 
we  raise  the  ball  of  a  pendulum  sideways  through  any  given 
elevation,  it  will  accumulate  so  much  power  or  momentum  in  its 
descent  through  the  arc  in  which  it  swings,  as  to  carry  it  up  to 
the  same  height  on  the  opposite  side  of  the  arc,  or  at  least  it  will 
do  so  nearly,  and  would  do  so  wholly  but  for  the  friction  of  the 
suspending  point  and  of  the  atmosphere,  which  will  cause  some 
slight  diminution  in  the  amount  of  elevation  at  each  successive 
beat.  If  a  hole  could  be  made  through  the  centre  of  the  earth, 
and  a  ball  were  suffered  to  drop  down  it,  the  velocity  would  go 
on  accelerating — supposing  there  were  no  resisting  atmosphere 
— until  the  centre  of  the  earth  were  reached;  and  the  ball  would 
then  pursue  its  course  with  a  velocity  gradually  diminishing  un- 
til it  reached  the  surface  at  the  antipodes,  when  it  would  come 
to  rest,  and  return — circulating  on  for  ever  from  surface  to  sur- 
face, in  a  manner  similar  to  that  in  which  a  pendulum  beats  hi 
its  arc.  If  we  suppose  an  atmosphere  to  be  introduced  into  the 
hole  or  tunnel,  then  the  ball  would  go  on  accelerating  only  until 
the  resistance  of  the  atmosphere  balanced  the  weight,  after 
which  no  further  acceleration  would  take  place.  This  is  the 
same  action  that  exists  when  a  railway-train  or  a  steam-vessel 
is  put  into  motion  by  an  engine.  In  each  case  the  train,  or 
steamer,  continues  to  accelerate  until  the  resistance  of  the  air 
or  of  the  water  balances  the  propelling  force,  after  which,  an 
equipoise  being  established,  no  further  acceleration  takes  place. 

The  velocity  which  bodies  acquire  by  falling  freely  by  gravity 
proceeds  according  to  a  known  law,  and  it  is  consequently  easy, 
when  we  know  the  height  from  which  a  body  has  fallen,  to  de- 
termine its  velocity ;  or  conversely,  when  we  know  its  velocity, 
we  can  easily  tell  from  what  height  it  must  have  descended. 


96 


MECHANICS    OF   THE    STEAM-ENGINE. 


Since,  too,  power  is  measurable  by  the  distance  through  which 
a  given  weight  is  lifted,  or  through  which  it  descends,  it  becomes 
easy  to  tell  when  we  know  the  weight  and  velocity  of  any  body, 
how  much  power  there  is  stored  up  in  it,  since  this  power  will, 
in  fact,  be  represented  by  the  weight  multiplied  by  the  height 
through  which  the  body  must  have  fallen  to  acquire  its  velocity. 
If  the  successive  additions  of  velocity  which  a  fallen  body 
receives  in  each  second  of  its  fall — namely,  32^  feet — be  repre- 
sented by  the  letter  #,  then  the  different  relations  of  the  time  of 
falling,  the  ultimate  velocity,  and  the  height  fallen  through,  will 
be  as  follows : — 


MOTION   OF  A  HEAVY  BODY  FALLISG  IN   VACTTO. 

0 
0 

0 
0 

1 

1(7 
If 
1? 

2 
2(7 
«? 

3? 

3 
8(7 
9? 
5? 

4 
40 
16? 
7? 

6 

5(7 
25? 
9? 

6 
6(7 
36? 

«? 

7 

10 
49? 
13? 

8 
8(7 
64? 
15? 

9 
9g 

81? 
17? 

10 

lOfir 
100? 
19? 

Ultimate  velocity  

Height  fallen  through  

Spaces  in  each  second.  

The  same  relations  are  shown  more  in  detail  in  the  following  table : 


MOTION  OF  A  BODY  FALLING  IN  VACTTO. 

Time  of  falling  in 
seconds. 

Height  fallen  in 
feet 

Velocity  acquired  in 
feet  per  second. 

0  rest 

1 

* 

0 

1-rk 
S-A 
9* 
16iV 

0 

8A 

16A 
24i 
32£ 

Ji 

H 

if 

2 

25tW 
36-ft 
49-Afe 
64£ 

40A 
48i- 
56-/4- 
64£ 

2i 
2i 
2* 
3 

Pitt 

lOOff 

rtftt 

144f 

72f 
8QA 

88H 
961. 

4 
5 
6 
7 
8 
9 

257i 
402£ 
579 
788£ 
10291 
1302| 

128$ 
160f 
193 
225£ 
257^ 
289£ 

LAWS   OP   FALLING   BODIES.  97 

RULES. 
VELOCITY  FROM  HEIGHT. 

TO   FIND    THE    VELOCITY    ACQTTIBED  BY  A  HEAVY  BODY  IK  FALL- 
ING THBOUGH  ANY  GIVEN  HEIGHT. 

RCLE. — Multiply  the  square  root  of  the  height  in  feet  through 
which  the  body  has  fallen  ly  the  constant  number  8-021.  The 
result  will  lie  the  velocity  in  feet  per  second  which,  the  body 
will  liave  attained. 

Example. — Suppose  a  leaden  bullet  to  be  dropped  from  a 
height  of  400  feet :  with  what  velocity  will  it  strike  the 
ground? 

Here  the  square  root  of  400  is  20,  and  20,  multiplied  by 
8-021-=160'42,  which  is  the  velocity  in  feet  per  second  which  the 
bullet  will  have  acquired  on  reaching  the  ground. 

The  same  result  is  attained  by  multiplying  the  space  fallen 
through  in  feet  by  64-333,  and  extracting  the  square  root  of  the 
product,  which  will  be  the  velocity  in  feet  per  second. 

VELOCITY   FROM   TIME. 

TO  FIND  THE  VELOCITY  IN  FEET  PEE  SECOND  WHICH  A  BODY 
WILL  ACQUIBE  BY  FALLING  FEEBLY  DUBING  ANY  GIVEN  NUM- 
BEB  OF  SECONDS. 

RULE. — Multiply  the  number  of  seconds  occupied  in  falling  "by 
32-166.  The  result  is  the  velocity  of  the  lody  in  feet  per 
second. 

Example. — Suppose  a  stone  to  be  dropped  from  such  a  height 
that  it  requires  four  seconds  to  reach  the  ground,  what  velocity 
will  the  stone  have  acquired  at  the  end  of  its  descent  f 

Here  four  seconds  multiplied  by  32-166=128-664,  which  is 
the  velocity  in  feet  per  second  that  the  stone  will  have  acquired 
on  reaching  the  ground. 
5 


98  MECHANICS   OF   THE    STEAM-ENGINE. 


HEIGHT   FROM   VELOCITY. 

TO  FIND  FEOM  THE  VELOCITY  ACQUIRED  BY  A  FALLING  BODY 
THE  HEIGHT  FROM  "WHICH  IT  MUST  HATE  FALLEN,  AND  ALSO 
THE  TIME  OF  THE  DESCENT. 

RULE. — Divide  the  square  of  the  acquired  velocity  in  feet  per 
second  by  64-333,  which  will  give  the  height  in  feet  from 
which  the  body  must  have  fallen  ;  and  divide  the  height  fallen 
by  the  constant  number  16*083,  and  extract  the  square  root 
of  the  quotient,  which  will  be  the  time  of  descent  in  seconds. 

Example. — If  a  stone  dropped  from  the  summit  of  a  tower 
strike  the  ground  with  a  velocity  of  120  feet  per  second,  what 
will  be  the  height  of  the  tower,  and  what  the  time  occupied 
by  the  stone  in  its  descent  ? 

Here  120  squared=14400  and  144400  divided  by  64-33  = 
223-84,  which  is  the  height  of  the  tower.  Further,  223-84 
divided  by  the  constant  number  16-083=13-9,  the  square  root  of 
which  is  3'72,  which  will  be  the  time  in  seconds  that  the  stone 
will  have  taken  to  fall  223-84  feet. 

HEIGHT   FROM  TIME. 

TO  FIND  FROM  THE  TIME  OCCUPIED  IN  THE  DESCENT  OF  A  FALL- 
ING BODY  WHAT  THE  HEIGHT  IS  FEOM  WHICH  IT  MUST  HAVE 
DESCENDED. 

RULE. — Multiply  the  square  of  the  time  occupied  in  the  descent 
in  seconds  by  the  constant  number  16-083.  The  product  is 
the  height  in  feet  from  which  the  body  must  have  fallen. 

Example. — If  a  stone  when  suffered  to  fall  into  a  well  strikes 
the  surface  of  the  water  in  four  seconds,  what  is  the  depth  of  the 
well  to  the  surface  of  the  water? 

Here  4  seconds  squared=16  seconds,  and  16  multiplied  by 
16-083=25Ti  feet,  which  is  the  depth  of  the  well  to  the  surface 
of  the  water. 


LAWS    OF   FALLING   BODIES.  99 


TIME   FROM  VELOCITY. 

TO  FIND  THE  TIME  IX  SECONDS  DUKING  WHICH  A  HEATY  BODY 
MUST  HAVE  CONTINUED  TO  FALL  TO  ATTAIN  ANT  GIVEN 
VELOCITY. 

RULE. — Divide  the  velocity  in  feet  per  second,  by  the  constant 
number  32-166.  The  quotient  is  the  number  of  seconds 
during  which  the  body  must  have  continued  to  fall  to  attain 
its  velocity. 

Example. — If  a  stone  in  falling  has  attained  a  velocity  on 
reaching  the  ground  of  128-664  feet  per  second,  how  many  sec- 
onds must  it  have  occupied  in  its  descent? 

Here  128-664  divided  by  32-166=4,  which  is  the  number  of 
seconds  that  the  stone  must  have  continued  to  fall  to  attain  its 
velocity. 

TIME  FROM  HEIGHT. 

TO     FIND     THE     TIME     IN    WHICH     A     HEAVY     BODY     WILL     FALL 
THROUGH   A    GIVEN    HEIGHT. 

RULE. — Divide  the  height  expressed  in  feet  by  the  constant  num- 
ber 16'083,  and  extract  the  square  root  of  the  quotient,  which 
will  give  the  time  in  seconds  in  which  the  heavy  body  will 
fall  through  the  given  height. 

Example. — Suppose  a  stone  to  be  let  fall  from  a  tower  400 
feet  high,  in  what  time  will  it  reach  the  ground  ? 

Here  400  divided  by  16-083=24-87,  and  the  square  root  of 
24'87  is  4-986,  or  very  nearly  5  seconds,  which  is  the  time  that 
would  elapse  before  the  stone  reached  the  ground. 

TO   FIND   THE  NUMBEE   OF   FEET   PASSED  THROUGH  BY   A   FALLING 
BODY   IN  ANY   GIVEN  SECOND   OF   ITS   DESCENT. 

RULE. — Multiply  the  number  of  the  second  by  32|  and  subtract 
from  the  product  16^.  The  remainder  will  be  the  number 
of  feet  passed  through  in  the  second  given. 


100  MECHANICS    OF   THE    STEAM-ENGINE. 

Example. — To  find  the  number  of  feet  passed  through  by  a 
falling  body  in  the  ninth  second  of  its  descent. 

Here  we  have  9  x32£=289£— 16^=273^,  which  is  the 
number  of  feet  passed  through  in  the  ninth  second  of  the  descent. 

MOTION  OF  FLUIDS. 

The  velocity  with  which  water  will  flow  out  of  a  hole  at  the 
side  or  in  the  bottom  of  a  cistern,  will  be  the  same  as  that  which 
a  heavy  body  will  acquire  in  falling  from  the  level  of  the  water 
surface  to  the  level  of  the  orifice,  and  may  easily  therefore  be 
computed  by  a  reference  to  the  laws  of  falling  bodies.  The 
atmosphere  exerts  a  pressure  of  about  14'7  Ibs.  per  square  inch, 
or  2116-4  Ibs.  per  square  foot,  on  all  bodies  on  the  earth's  sur- 
face ;  and  if  the  atmosphere  be  pumped  out  of  the  space  beneath 
a  piston,  while  suffered  to  press  on  its  upper  surface,  the  piston 
will  be  forced  downward  in  its  cylinder  with  a  pressure  of  14'7 
Ibs.  on  each  square  inch  of  the  piston's  area.  In  a  common 
sucking  pump  the  water  is  drawn  up  after  the  piston,  in  conse- 
quence of  the  production  of  a  partial  vacuum  beneath  the  piston ; 
and  the  water  in  the  well  being  subjected  to  the  pressure  of  the 
atmosphere  while  the  pressure  is  removed  from  the  water  in  the 
pump  barrel,  the  water  rises  in  the  suction  pipe,  and  would  con- 
tinue to  do  so  if  the  pump  were  raised  further  and  further  up, 
until  a  column  of  water  had  been  interposed  between  the  pump- 
barrel  and  the  well  sufficiently  high  to  balance  the  weight  of  the 
atmosphere.  The  water  will  cease  to  rise  any  higher  after  this 
altitude  has  been  attained. 

"When  we  know  the  weight  of  a  cubic  inch  or  cubic  foot  of 
•water,  it  is  easy  to  tell  the  number  of  cubic  inches  or  cubic  feet 
that  must  be  piled  upon  one  another  to  produce  a  weight  of 
14*7  Ibs.  on  the  square  inch  or  211 6'4  Ibs.  on  the  square  foot; 
and  it  will  be  found  to  be  408  cubic  inches  in  the  case  of  the 
cubic  inches,  or  a  column  1  inch  square  and  34  feet  high,  or  84 
cubic  feet  in  the  case  of  the  cubic  feet.  Mercury  being  about 
13'6  times  heavier  than  water,  a  column  of  mercury  1  inch 
square  and  30  inches  high  will  weigh  about  15  Ibs.  A  column 


MOTION   AND   WEIGHT   OF  FLUIDS.  101 

of  air  high  enough  to  weigh  15  Ibs.,  will  be  773'29  times  higher 
than  a  column  of  water  of  the  same  weight — water  being  773-29 
times  heavier  than  air  at  the  ordinary  barometric  density  of  29-9 
inches  of  mercury.  In  other  words,  the  height  of  a  column  of 
air  1  inch  square  and  the  same  density  as  that  on  the  earth's 
surface,  that  will  weigh  15  Ibs.,  will  be  34x773-29  =  25521-86 
feet,  or  taking  the  atmospheric  pressure  at  14-7  Ibs.,  the  height 
will  be  26214  feet.  The  velocity  therefore  with  which  water 
will  rush  into  a  vacuum,  will  be  equal  to  that  which  a  heavy 
body  will  acquire  in  falling  through  a  height  of  34  feet.  The 
velocity  with  which  mercury  will  flow  into  a  vacuum,  will  be 
equal  to  that  which  a  heavy  body  will  acquire  by  falling  through 
a  height  of  2£  feet ;  and  the  velocity  with  which  air  will  flow 
into  a  vacuum,  will  be  equal  to  that  which  a  heavy  body  will 
acquire  by  falling  through  a  height  of  26214  feet.  Now  the 
velocity  which  a  heavy  body  will  acquire  in  falling  through  34 
feet  will  be  equal  to  the  square  root  of  34,  which  is  5*8  multi- 
plied by  the  constant  number  8-021 ;  or  it  will  be  46-5218  feet 
per  second,  which  consequently  will  be  the  velocity  with  which 
water  will  flow  into  a  vacuum.  The  velocity  with  which  mer- 
cury will  flow  into  a  vacuum  will  be  12-83  feet  per  second, 
for  the  square  root  of  2-J  is  1'6  nearly,  and  1-6  multiplied  by 
8-021  — 12-8336.  The  velocity  with  which  air  weighing  0*080728 
Ibs.  per  cubic  foot  will  flow  into  a  vacuum  will  be  1298.5999 
feet  per  second ;  for  the  square  root  of  26214  is  161'9  nearly, 
which  multiplied  by  8-021  — 1298-5999  feet  per  second.  The 
density  of  the  air  here  supposed  is  the  density  at  the  tempera- 
ture of  melting  ice.  At  the  ordinary  atmospheric  temperatures 
the  density  will  be  somewhat  less ;  and  if  the  density  be  taken 
so  that  the  height  of  the  homogeneous  atmosphere,  as  it  is 
called,  or  of  that  imaginary  atmosphere  which  produces  the 
pressure — and  which  is  supposed  to  be  of  uniform  density 
throughout  its  depth — is  27,818  feet,  then  the  velocity  of  the  air 
rushing  into  a  vacuum  will  be  a  little  greater  than  what  it  has 
been  here  reckoned  at,  or  it  will  be  1338  feet  per  second. 
These  velocities  it  will  be  understood  are  the  theoretical  veloci- 
ties, which  can  in  no  case  be  exceeded ;  but  which  are  fallen 


102  MECHANICS    OF   THE    STEAM-ENGINE. 

short  of  in  practice  to  a  greater  or  less  extent,  depending  on  the 
size  and  form  of  the  orifice  through  which  the  air  enters,  and 
other  analogous  circumstances. 

The  velocity  with  which  steam  or  any  vapour  or  gas  what- 
ever will  rush  into  a  vacuum,  can  easily  be  determined  when 
we  know  its  pressure  and  density ;  for  taking  into  account  the 
density,  or  the  weight  of  one  cubic  foot,  we  have  merely  to  see 
how  many  of  these  cubic  feet  must  be  piled  upon  one  another 
to  produce  the  given  pressure  or  weight  upon  the  square  foot  of 
base ;  and  the  velocity  will  be  in  every  case  the  same  as  that 
which  a  heavy  body  would  acquire  in  falling  through  the  height 
of  the  column  required  to  produce  the  weight.  Thus  it  is  found 
that  the  density  of  steam  of  the  atmospheric  pressure  is  about 
1700  times  less  dense  than  water.  Mr.  Watt  reckoned  that  a 
cubic  inch  of  water  produced  a  cubic  foot  or  1728  cubic  inches 
of  steam,  having  the  same  pressure  as  the  atmosphere ;  and  if 
the  pressure  of  the  atmosphere  be  equal  to  the  pressure  pro- 
duced by  34  feet  of  water,  then,  if  we  reckon  steam  as  1700 
tunes  less  dense  than  water,  it  would  require  1700  columns  of 
steam,  each  34  feet  high,  placed  on  top  of  one  another,  to  exert 
the  same  weight  or  pressure  as  one  column  of  water  34  feet 
high.  Now  1700  hundred  times  34  is  57800,  which  therefore  is 
the  height  a  column  of  steam  1700  times  less  dense  than  water 
would  require  to  have  in  order  to  balance  the  pressure  of  the 
atmosphere  or  of  34  feet  of  water.  The  velocity  which  a  body 
would  acquire  in  falling  through  a  height  of  57800  feet,  is  1926-6 
feet  per  second ;  for  the  square  root  of  57800  is  240-2  nearly, 
and  240-2  multiplied  by  8-021=1926-6442  feet  per  second,  which 
is  consequently  the  velocity  with  which  steam  of  this  pressure 
would  rush  into  a  vacuum.  The  velocity  with  which  steam  of 
a  greater  pressure  than  that  of  the  atmosphere  will  rush  into  a 
vacuum,  will  not  be  sensibly  greater  than  that  of  steam  of  the 
atmospheric  pressure.  For  as  the  density  of  the  steam  increases 
in  nearly  the  same  ratio  as  its  pressure,  the  column  will  require 
to  be  as  much  lower,  by  virtue  of  the  increased  density,  as  it 
requires  to  be  higher  to  give  the  increased  pressure.  In  other 
words,  the  height  of  the  theoretical  column  of  steam  required 


103 

to  produce  the  pressure,  will  be  nearly  the  same  at  all  pressures ; 
since  a  low  column  of  dense  steam  will  produce  the  same  press- 
ure as  a  high  column  of  rare,  and  the  density  and  pressure  ad- 
vance in  nearly  the  same  ratio.  It  may  hence  be  concluded  that 
steam  of  all  pressures  will  rush  into  a  vacuum  with  a  velocity 
of  about  2,000  feet  per  second,  if  the  vacuum  be  perfect  and  the 
flow  unimpeded. 

If  steam,  instead  of  being  suffered  to  escape  into  a  vacuum, 
be  made  to  issue  into  a  vessel  containing  steam  of  a  lower  press- 
ure, the  velocity  of  efflux  will  be  the  same  as  that  which  a 
heavy  body  would  acquire  in  falling  from  the  top  of  the  column 
of  steam  required  to  produce  the  greater  pressure,  to  the  top  of 
a  lower  column  of  the  same  steam  adequate  to  produce  the 
lesser  pressure.  Thus  if  we  have  steam  with  a  pressure  of  two 
atmospheres,  flowing  into  steam  with  a  pressure  of  one  atmos- 
phere, then,  inasmuch  as  the  density  or  weight  of  the  steam 
increases  very  nearly  in  the  same  proportion  as  its  pressure,  a 
cubic  inch  of  steam  with  a  pressure  of  two  atmospheres  will  be 
about  twice  as  heavy  as  a  cubic  inch  of  steam  with  a  pressure 
of  one  atmosphere.  Such  steam,  therefore,  instead  of  being 
1700  times  less  dense  than  water,  will  be  the  half  of  this  or  only 
850  times  less  dense  than  water.  A  column  of  this  steam, 
therefore,  850  times  34  feet=28900  feet  high,  will  exert  a  press- 
ure of  one  atmosphere,  or  about  15  Ibs.  on  each  square  inch ; 
and  a  column  of  twice  this  height,  or  57800  feet,  will  exert  a 
pressure  of  two  atmospheres  or  80  Ibs.  on  each  square  inch. 
The  velocity  with  which  the  steam  will  rush  from  one  vessel  to 
the  other,  will  be  the  same  as  that  which  a  heavy  body  would 
acquire  in  falling  from  the  height  of  the  column  of  the  denser 
steam  required  to  produce  the  higher  pressure  to  the  top  of  the 
column  of  the  same  steam  of  such  height  as  would  produce  the 
less  pressure ;  and  as  in  this  case  the  heights  of  such  columns 
will  be  1700  x  34  feet,  and  850  x  34  feet,  or  57800  and  28900  feet, 
the  difference  of  height  will  be  28900  feet ;  and  the  velocity  of 
efflux  from  one  vessel  into  the  other  will  be  equal  to  that  which 
a  heavy  body  would  acquire  by  falling  through  a  height  of 
28900  feet.  Now  the  square  root  of  28900  is  170 ;  and  170 


104 


MECHANICS   OF  THE   STEAM-ENGINE. 


multiplied  by  8>021=1363<5Y  feet  per  second,  which  is  the  ve- 
locity with  which  steam  with  a  pressure  of  two  atmospheres 
would  rash  into  steam  with  a  pressure  of  one  atmosphere.  This 
consequently  may  be  reckoned  as  the  velocity  with  which  steam 
of  15  Ibs.  pressure  above  the  atmosphere  would  rush  into  the 
atmosphere.  Such  velocities  at  different  pressures  are  exhibited 
in  the  following  table : — 


VELOCITY   OF  EFFLUX   OF  HIGn-PEESSTTEE   STEAM   INTO   THE 
ATMOSPHERE. 


Pressure  of 
steam  above  the 
atmosphere. 

Velocity  of  free  | 
efflux  in  feet  per 
second. 

Pressure  of 
steam  above  the 
atmosphere. 

Velocity  of  free 
efflux  in  feet  per 
second. 

Ibs. 

feet. 

Ibs. 

feet. 

1 

482 

50 

1791 

2 

663 

60 

1838 

3 

791 

70 

1877 

4 

890 

80 

1919 

5 

973 

90 

1936 

10 

1241 

100 

1957 

20 

1604 

110 

1972 

80 

1643 

120 

1990 

40 

1729 

130 

2004 

This  table  is  computed  by  taking  the  difference  of  the  two 
pressures  for  the  effective  pressure,  which  effective  pressure  is 
expressed  in  pounds  per  square  inch,  divided  by  the  weight  of  a 
cubic  foot  of  the  denser  fluid  in  pounds,  and  the  square  root  of 
the  quotient  is  multiplied  by  96.  The  denser  the  fluids  are  the 
less,  it  is  clear,  will  be  the  velocity  of  efflux  which  a  given  differ- 
ence of  pressure  will  create ;  for  the  heights  of  the  columns,  and 
also  the  difference  of  their  heights,  will  be  small  in  the  propor- 
tion of  the  density  of  the  denser  fluid.  The  more  dense  the 
fluid  is,  the  larger  becomes  the  mass  of  matter  which  a  given 
pressure  has  to  move.  With  steam  of  16  Ibs.  pressure  flowing 
into  steam  or  air  of  15  Ibs.  pressure,  the  moving  pressure  is  1  lb., 
and  the  velocity  of  efflux  is  482  feet  per  second.  With  steam 
of  101  Ibs.  pressure  flowing  into  steam  or  air  of  100  Ibs.  pressure, 


INERTIA   AND   MOMENTUM.  105 

the  moving  pressure  is  the  same,  but  the  velocity  of  efflux  will 
only  be  207  feet  per  second. 

INERTIA  AND  MOMENTUM. 

"When  a  body  is  moved  from  a  state  of  rest  to  a  state  of  mo- 
tion, or  from  a  slow  motion  to  a  faster,  power  is  absorbed  by 
the  body ;  and  when  a  body  is  brought  from  a  state  of  motion 
to  rest,  or  from  a  fast  motion  to  a  slow  one,  power  is  liberated 
by  the  body.  The  quality  which  enables  a  body  to  resist  the 
sudden  communication  of  motion  is  termed  its  Inertia ;  and  the 
quality  which  enables  a  body  to  resist  the  sudden  extinction  of 
motion  is  termed  its  Momentum.  Whatever  power  a  body  ab- 
sorbs in  being  put  into  motion,  it  afterwards  surrenders  in  being 
brought  to  a  state  of  rest ;  and  the  amount  of  power  existing  in 
any  moving  body  is  measurable  by  its  weight  multiplied  by  the 
square  of  its  velocity,  or  by  the  height  through  which  it  must 
have  fallen  by  gravity  to  attain  its  velocity. 

A  railway  carriage  of  ten  tons'  weight,  therefore,  moving  at 
a  speed  of  20  miles  an  hour,  will  have  as  great  a  momentum  as 
4  railway  carriages  weighing  10  tons  each  moving  at  the  rate  of 
10  miles  an  hour.  In  like  manner  the  momentum  of  a  cannon 
ball  moving  at  a  velocity  of  1,700  feet  a  second,  will  be  28,900 
times  greater  than  if  it  moved  at  a  speed  of  10  feet  per  second, 
since  the  square  of  1,700  is  to  the  square  of  10  as  28,900  to  1. 
Josephus  mentions  that  some  of  the  battering-rams  employed  by 
the  Romans  in  Judea  were  90  feet  long,  and  weighed  1,500  tal- 
ents of  114  Ibs.  to  the  talent,  or  76-3392  tons.  The  weight  of  a 
cannon  ball  which  has  the  same  amount  of  mechanical  power 
stored  up  in  it,  or  which  will  give  the  same  force  of  impact 
when  moving  at  a  speed  of  1,800  feet  per  second,  as  the  batter- 
ing-ram will  do  when  moving  at  a  velocity  of  10  feet  per  second, 
can  easily  be  determined ;  for  we  have  only  to  multiply  76*3392 
tons  by  the  square  of  10  and  divide  by  the  square  of  1,800, 
which  will  give  '0023561  tons,  or  5-12776  Ibs.,  as  the  weight  of 
the  ball  required. 
5* 


106  MECHANICS    OF   THE    STEAM-EXGINE. 

TO  FIND  THE  QUANTITY  OF  MECHANICAL  POWER  REQUIRED  TO  COM- 
MUNICATE DIFFERENT  VELOCITIES  OF  MOTION  TO  HEAVY  BODIES. 

EULE. — Multiply  the  mass  of  matter  fiy  the  height  due  to  the 
velocity  it  has  acquired,  supposing  that  it  attained  its  ve- 
locity ly  falling  by  gravity.  The  product  is  the  mechanical 
power  communicated  in  generating  that  velocity  of  motion 
in  the  body. 

Example  1. — Suppose  a  waggon  on  a  railway  to  weigh  2,500 
pounds,  what  mechanical  power  must  be  communicated  to  it  to 
urge  it  from  rest  into  motion  with  a  velocity  of  3  miles  an  hour, 
or  4-4  feet  per  second  ? 

Now  here  the  height  in  feet  from  which  a  hody  must  have 
fallen  to  acquire  any  given  velocity  will  be  the  square  of  the 
velocity  in  feet  per  second  divided  by  64J ;  or  it  will  be  the 
square  of  the  quotient  obtained  by  dividing  the  velocity  in  feet 
per  second  by  the  square  root  of  64J,  or  8'021.  Now  4 -4  -f-  8-021 
=  •5487,  the  square  of  which  is  '301  feet,  the  height  that  a  body 
must  fall  to  acquire  a  velocity  of  3  miles  an  hour.  Hence  the 
mechanical  power  communicated  is  2,500  Ibs.  x  -301  ft.  —  752*5 
Ibs.  descending  through  1  foot. 

Example  2. — Required  the  mechanical  effect  treasured  up  in 
a  cast-iron  fly-wheel,  the  mean  diameter  of  which  is  30  feet  with 
a  sectional  area  of  rim  of  60  square  inches,  and  making  20  turns 
in  the  minute. 

The  diameter  of  the  wheel  being  30  feet,  the  circumference 
will  be  94*248  feet,  and,  as  the  wheel  makes  20  revolutions  in 
the  minute,  the  velocity  of  the  rim  will  be  94-248  x  20  =  1884-96 
feet  per  minute,  or  31*416  feet  per  second.  Again  the  cubical 
content  of  the  rim  in  cubic  feet  being  60  x94'248-*-144  =  39*27 
cubic  feet,  and  the  weight  of  a  cubic  foot  of  cast-iron  being  45  3£ 
Ibs.,  we  have  39*27x453^=17794-22  Ibs.  as  the  weight  of  the 
rim.  Hence  the  mechanical  effect  treasured  up  in  the  rim  of 
this  wheel  is  17794-22  x(31*416-*-8*021)2=268,650  Ibs.  raised 
one  foot  high.  This  it  will  be  observed  is  about  eight  actual 
horse-power.  The  mechanical  energy  with  which  the  fly-wheel 
of  an  engine  is  generally  endowed,  is  equal  to  the  power  exerted 


BODIES  REVOLVING  IN  A  CHICLE— CENTRIFUGAL  FORCE.  107 

in  from  four  to  six  half  strokes  of  the  engine,  or  two  to  three 
complete  revolutions ;  so  that  the  fly-wheel  above  particularized 
is  such  as  would  be  suitable  for  an  engine  which  exerts  a  power 
of  four  actual  horses,  or  four  times  33,000  pounds  raised  one 
foot  high  in  each  revolution,  or  80  horses'  power. 

BODIES  REVOLVING  IN  A  CIRCLE. 

When  bodies  revolve  in  circles  round  fixed  axes  of  motion, 
the  different  particles  can  have  no  motion  except  in  circles  de- 
scribed round  such  fixed  axes ;  and  the  velocities  of  the  particles 
composing  the  body  must  be  greater  or  less,  depending  upon 
their  distance  from  the  centre  round  which  the  body  revolves. 
To  apply  the  laws  of  falling  bodies  to  this  case  we  must  imagine 
the  particles  composing  such  revolving  bodies  to  be  divided  and 
collected  into  several  small  bodies  situated  at  different  distances 
from  the  centre,  and  therefore  moving  with  different  velocities ; 
and  then  we  may  determine  the  power  which  must  be  commu- 
nicated to  each  of  the  supposed  separate  bodies  to  give  it  the  ve- 
locity which  it  actually  possesses.  The  sum  of  all  the  powers 
so  determined  is  the  total  power  which  must  be  communicated 
to  the  body,  to  give  to  it  the  velocity  of  motion  with  which  it 
actually  revolves.  Thus  a  rod  moving  about  one  of  its  extrem- 
ities may  be  supposed  to  be  compounded  of  a  number  of  balls, 
like  a  string  of  beads  strung  on  a  wire.  The  velocity  of  each 
of  these  balls  can  then  be  ascertained,  which  will  enable  us  to 
compute  the  mechanical  power  resident  in  it,  and  which  will  be 
the  same  as  if  it  moved  in  a  straight  line.  The  sum  of  the  quan- 
tities thus  ascertained  will  be  the  total  mechanical  power  resi- 
dent in  the  revolving  body. 

CENTRIFUGAL  FORCE. 

The  centrifugal  force  of  a  body  which  revolves  in  any  circle 
in  a  given  time,  is  proportional  to  the  diameter  of  the  circle  in 
which  it  revolves.  Thus,  in  the  case  of  two  fly-wheels  of  the 
same  weight  but  one  of  twice  the  diameter  of  the  other,  the 


108  MECHANICS    OF   THE    STEAM-ENGINE. 

larger  wheel  will  have  twice  the  amount  of  centrifugal  force 
that  the  small  one  has. 

The  centrifugal  force  of  a  body  moving  with  different  veloc- 
ities in  the  same  circle  is  proportional  to  the  square  of  the 
velocities  with  which  it  moves  in  that  circle ;  or,  what  is  the 
same  thing,  to  the  square  of  the  number  of  revolutions  per- 
formed in  a  given  time.  Thus,  the  fly-wheel  of  any  engine  will 
have  four  times  the  amount  of  centrifugal  force  it  possessed 
before,  if  driven  at  twice  the  speed.  In  Mr.  Watt's  engines 
with  sun  and  planet  wheels,  in  which  the  fly-wheel  made  twice 
the  number  of  revolutions  made  by  the  engine,  the  fly-wheel 
had  four  times  the  centrifugal  force  that  would  be  possessed  by 
the  same  fly-wheel  if  coupled  immediately  to  the  crank. 

The  centrifugal  force  of  a  body  of  a  given  weight,  revolving 
with  a  certain  uniform  velocity  in  a  circle  of  a  given  diameter, 
was  investigated  by  the  Marquis  de  1'Hopital,  who  gave  the  rule 
for  ascertaining  this  force  that  is  now  generally  followed.  It  is 
founded  on  the  consideration  of  the  height  from  which  the  body 
must  have  fallen  by  gravity  to  have  acquired  the  velocity  with 
which  its  centre  of  gyration  moves  in  the  circle  which  it  de- 
scribes. Then  as  the  radius  of  that  circle  is  to  double  the  height 
due  to  the  velocity,  so  is  the  weight  of  the  body  to  its  centrif- 
ugal force. 

TO  FIND  THE  CENTRIFUGAL  FOEOE  OF  A  BODY  OF  A  GIVEN  WEIGHT 
EEVOLVING  IN  A  CIEOLE  OF  A  GIVEN  DIAMETER. 

EULE. — Divide  tJie  velocity  in  feet  per  second,  ~by  4'01,  and,  the 
square  of  the  quotient  is  four  times  the  height  in  feet  due  to 
the  velocity.  Divide  this  quadrupled  height  by  the  diameter 
of  the  circle,  and  the  quotient  is  the  centrifugal  force  when 
the  weight  of  the  tody  is  1  /  consequently,  multiplying  it  by 
the  weight  of  the  body  gives  the  actual  centrifugal  force  in 
pounds  or  tons. 

Example  1. — Suppose  that  the  rim  of  a  fly-wheel  30  feet  di- 
ameter and  weighing  15718  Ibs.,  moves  at  the  rate  of  27*49  feet 
per  second,  what  will  be  its  centrifugal  force  ?  Here  we  have 


CENTRIFUGAL    FORCE    OF   FLY-WHEELS.  109 

the  velocity  27-49-j-4-01=6'85,  which,  squared,  is  46'9225  ;  and 
this,  divided  by  30,  is  1/564 :  so  that  the  centrifugal  force  is  1'564 
times  the  weight  of  the  body,  or  10'97  tons. 

Example  2. — Suppose  that  the  rim  of  a  fly-wheel  which  is  20 
feet  diameter  moves  with  a  velocity  of  82£  feet  per  second :  then 
32-16-5-4-01=8-02,  the  square  of  which  is  64'32  feet,  which  is 
the  quadrupled  height  due  to  the  velocity,  and  this  divided  by 
20  feet  diameter  gives  3'216  tunes  the  weight  of  the  rim  as  the 
centrifugal  force. 

ANOTHER  ETJLE. — Multiply  the  square  of  the  number  of  revolu- 
tions per  minute  T)y  the  diameter  of  the  circle  of  revolution 
in  feet,  and  divide  the  product  by  the  constant  number  5870; 
the  quotient  is  the  centrifugal  force  of  the  body  in  terms  of 
its  weight,  which  is  supposed  to  be  1. 

Example  1. — Suppose  a  stone  of  2  Ibs.  weight  is  placed  in  a 
sling,  and  whirled  round  in  a  circle  of  4  feet  diameter,  at  the 
rate  of  120  revolutions  per  minute :  then  120  squared=14400  x  4 
feet  diameter=57600->5870=9'81  which  is  the  ratio  of  the  cen- 
trifugal force  to  the  weight ;  and,  the  weight  being  2  Ibs.,  the 
centrifugal  force  acting  to  break  the  string  and  escape  is  19 -6  Ibs. 
Example  2. — In  the  case  of  the  first  fly-wheel  30  feet  diam- 
eter, referred  to  above,  we  multiply  the  square  of  the  number 
of  revolutions  per  minute  (1T&)  by  the  diameter  of  the  circle  in 
feet  (30),  and  divide  the  product  by  5870 ;  which  gives  the  cen- 
trifugal force  in  terms  of  the  weight  of  the  body,  and  17J2  x  30 
-=-5870=1-564  as  before. 

TO  FIND  THE  BATE  AT  WHICH  A  BODY  MUST  REVOLVE  IN  ANT  OIB- 
OLE,  THAT  ITS  CEBTrBIffUGAL  FOBOE  MAY  BE  EQUAL  TO  ITS 
WEIGHT. 

KULE. — Divide  the  constant  number  5870  by  the  diameter  of  the 
circle  in  feet,  and  the  square  root  of  the  quotient  is  the  num- 
ber of  revolutions  it  will  make  per  minute,  when  the  centrif- 
ugal force  is  equal  to  the  weight. 

Example. — In  a  circle  of  6'5  feet  diameter,  a  body  must  re- 
volve about  30  times  a  minute  that  its  centrifugal  force  may  be 


110  MECHANICS    OF   THE    STEAM-ENGINE. 

equal  to  its  weight;   for  5870^-6*5  =  903,  the  square  root  of 
which  is  30'05  revolutions  per  minute. 

The  mechanical  power  which  must  be  communicated  to  a 
solid  disc  of  uniform  density,  to  make  it  revolve  on  its  axis,  is 
the  same  as  that  which  must  be  communicated  to  one-half  of  its 
weight  of  matter,  to  give  it  motion  in  a  straight  line  with  the 
same  velocity  with  which  the  circumference  of  the  disc  moves 
in  a  circle. 

TO   DETERMINE   THE  BURSTING   STRAIN   OF  A   FLY-WHEEL. 

If  we  suppose  half  of  a  fly-wheel  to  be  securely  attached  to 
the  axis,  while  the  other  half  is  held  only  by  the  rim  or  by  bolts 
which  it  tends  to  break  by  its  centrifugal  force,  then  there  will 
be  a  velocity  at  which  the  centrifugal  force  of  half  the  rim  will 
overcome  the  cohesion  of  the  metal  of  the  rim,  or  of  the  bolts, 
and  the  wheel  will  be  burst  by  its  centrifugal  force. 

In  mechanical  works  it  has  been  usual  to  reckon  the  cohesive 
strength  of  wr ought-iron  within  the  limits  of  elasticity  at  17,800 
Ibs.  per  square  inch  of  section,  and  of  cast-iron  at  15,300  Ibs. 
per  square  inch  of  section ;  by  which  is  meant  that  a  bar  of 
wrought-iron  one  inch  square  might  be  stretched  by  a  weight 
of  17,800  Ibs.  without  injury,  and  a  bar  of  cast-iron  might  be 
stretched  by  a  weight  of  15,300  Ibs.  without  injury,  and  though 
somewhat  drawn  out  by  such  weights,  would,  like  a  spiral 
spring,  again  return  to  the  original  length  on  the  weight  being 
removed.  This  estimate  for  cast-iron  is  much  too  high ;  and  in 
machinery  wrought-iron  should  not  be  loaded  with  more  than 
4,000  Ibs.  per  square  inch  of  section,  and  cast-iron  should  not  be 
loaded  with  more  than  2,000  Ibs.  per  square  inch  of  section. 
The  breaking  tensile  strength  of  good  wrought-iron  is  about 
60,000  Ibs.  per  square  inch  of  section,  and  of  good  cast-iron 
about  15,000  Ibs.  per  square  inch  of  section.  But  both  wrought 
and  cast-iron  will  be  broken  gradually  with  much  less  strain 
than  would  be  required  to  break  them  at  once ;  and  if  the  limit 
of  elasticity  be  exceeded,  they  will  undergo  a  gradual  deteri- 
oration, and  will  be  broken  in  the  course  of  time.  If  the  velocity 


POWER   IN    A   REVOLVING   DISC.  Ill 

of  rotation  of  a  cast-iron  fly-wheel  be  so  great  that  its  centrif- 
ugal force  becomes  greater  than  15,000  Ibs.  in  each  square  inch 
of  the  section  of  the  rim,  it  will  necessarily  burst,  as  a  wrought- 
iron  one  would  also  do  if  the  centrifugal  force  exceeded  60,000 
Ibs.  per  square  inch  of  section.  But  to  be  within  the  limits  of 
safety,  a  strain  of  4,000  Ibs.  per  square  inch  of  section  should 
not  be  exceeded  for  wrought-iron,  and  2,000  Ibs.  per  square  inch 
of  section  for  cast. 

TO   DETERMINE   THE   MECHANICAL  POWER   RESIDENT  IN  A  EE- 
VOLVING  DISC. 

RULE. — Multiply  one-half  of  the  weight  of  the  revolving  disc 
by  the  height  due  to  the  velocity  with  which  the  circum- 
ference of  the  icheel  or  disc  moves  ;  the  product  is  the  me- 
chanical power  communicated. 

Example  1. — Suppose  that  a  grindstone  4'375  feet  diameter, 
weighing  3,500  Ibs.,  makes  270  revolutions  per  minute;  what 
power  must  be  communicated  to  it  to  give  it  that  motion  ? 

The  velocity  of  the  circumference  will  be  61'83  feet  per 
second,  and  the  height  due  to  this  velocity  is  59 '4  feet.  The 
mechanical  power  is  1,750  Ibs.  (half  the  weight)  x  59*4  feet  = 
103'950  Ibs.  raised  one  foot. 

If  the  revolving- wheel  is  not  an  entire  disc  or  solid  circle,  but 
only  a  ring  or  annulus,  it  must  first  be  considered  as  a  disc,  and 
the  effect  of  the  part  which  is  wanting  must  then  be  calculated 
and  deducted. 

Example  2. — Suppose  the  rim  of  a  cast-iron  fly-wheel  to  be 
22  feet  diameter  outside,  and  20  feet  inside,  and  that  the  thick- 
ness of  the  rim  is  6  inches,  and  that  the  wheel  makes  36  revo- 
lutions per  minute,  what  power  must  be  communicated  to  the 
rim  to  give  it  that  motion,  the  weight  of  the  arms  being  left  out 
of  the  account? 

A  solid  wheel  22  feet  diameter  and  6  inches  thick  would 
contain  190  cubic  feet,  from  which,  if  we  deduct  157  cubic  feet, 
which  would  be  the  capacity  of  a  solid  wheel  20  feet  diameter 
and  6  inches  thick,  we  have  33  cubic  feet  as  the  cubical  contents 


112  MECHANICS    OF   THE    STEAM-ENGINE. 

of  the  annulus.  Now  in  the  case  of  a  solid  wheel  of  22  feet 
diameter,  the  velocity  of  the  circumference  at  36  revolutions 
per  minute  would  be  41 '47  feet  per  second,  the  height  due  to 
which  would  be  26'8  feet,  which  multiplied  by  95  cubic  feet  (or 
half  the  mass)  gives  2,546  cubic  feet  of  cast-iron,  raised  1  foot  for 
the  power  communicated.  Then  supposing  another  solid  wheel 
20  feet  diameter,  we  shall  find  by  a  like  mode  of  computation 
that  the  power  communicated  is  equivalent  to  1,735  cubic-feet 
of  cast-iron  raised  through  1  foot.  This  deducted  from  2,546 
leaves  811  cubic  feet  raised  through  1  foot  as  the  power  resident 
in  the  annulus ;  and  if  we  take  the  weight  of  a  cubic  foot  of 
cast-iron  in  round  numbers  as  480  Ibs.,  we  have  389,280  Ibs. 
raised  1  foot,  for  the  mechanical  power  which  must  be  commu- 
nicated to  the  rim  of  the  fly-wheel  in  question  to  give  it  a  ve- 
locity of  36  revolutions  per  minute. 

The  mechanical  power  which  must  be  communicated  to  solid 
discs  of  different  diameters,  but  of  the  same  thickness  and  den- 
sity, to  make  them  revolve  in  the  same  time,  is  as  the  fourth 
powers  of  their  diameters. 

CENTRES  OF  GYRATION  AND  PERCUSSION. 

The  centre  of  gyration  is  a  point  in  bodies  which  revolve  in 
circles  in  which  the  momentum,  or  energy  of  the  moving  mass, 
may  be  supposed  to  be  collected.  It  is  in  the  same  point  as  the 
centre  of  percussion  of  revolving  bodies,  because  a  revolving 
body,  if  suffered  to  strike  another  body  that  is  either  at  rest  or 
that  moves  with  a  different  velocity  in  the  same  orbit,  will 
neither  be  deflected  to  the  right  nor  to  the  left,  but  will  act  just 
as  if  the  whole  mass  of  matter  were  collected  in  that  point.  In 
bodies  moving  forward  in  a  straight  line,  the  centre  of  percus- 
sion is  in  the  centre  of  gravity ;  but,  in  bodies  revolving  in  cir- 
cles, the  part  of  the  body  most  remote  from  the  centre  of  the 
circle  moves  with  a  different  velocity  from  the  part  nearest  to 
the  centre  of  the  circle.  The  centre  of  percussion,  therefore, 
cannot  be  in  the  centre  of  gravity  in  such  a  case,  but  at  some 
point  nearer  the  circumference  of  the  circle  ;  and  the  line  traced 


TO   FIND   THE   CENTRE   OP   GTKATION.  113 

by  that  point  will  divide  the  body  into  two  parts,  each  having 
the  same  amount  of  mechanical  power  treasured  in  them,  or 
each  requiring  the  same  amount  of  mechanical  power  to  put 
them  into  revolution  at  their  existing  velocity.  If  the  body, 
therefore,  could  be  divided  instantly,  and  without  violence, 
through  the  line  traced  by  the  centre  of  gyration,  each  portion 
of  the  body  would  continue  to  revolve  with  its  former  velocity. 
The  point  tracing  the  line  which  thus  divides  the  body  is  the 
centre  of  percussion,  and  also  the  centre  of  gyration,  and  in  re- 
volving bodies  these  centres  are  identical.  If  a  given  pressure 
act,  through  a  given  space,  upon  a  body  at  its  centre  of  gyration, 
in  the  direction  of  a  tangent  to  the  circle  which  that  centre  must 
describe  round  the  fixed  centre  of  motion,  such  an  amount  of 
power  will  move  the  centre  of  gyration  with  the  same  velocity 
in  its  circle  of  revolution,  as  it  would  move  an  equal  mass  of 
matter  in  a  right  line  by  acting  at  the  centre  of  gravity  of  the 
mass.  If  the  whole  mass  of  the  revolving  body  could  be  col- 
lected into  its  centre  of  gyration,  the  mechanical  power  resident 
in  the  body  would  be  represented  by  multiplying  the  total 
weight  of  the  body  by  the  square  of  the  velocity  of  the  centre 
of  gyration. 

TO  FIND    THE    DISTANCE    OF    THE  CENTRE    OF  GYRATION    OF  ANT 
REVOLVING  BODY  FROM   THE   CENTRE   OR  AXIS   OF   MOTION. 

EULE. — Multiply  the  weight  of  each  particle,  or  equal  small 
portion  of  the  body,  ~by  the  square  of  its  distance  from  the 
axis,  and  divide  the  sum  of  all  these  products  <by  the  weight 
of  the  whole  mass;  the  square  root  of  the  quotient  will  le 
the  distance  of  the  centre  of  gyration  from  the  axis  of  motion. 

Example. — Suppose  three  cannon  balls  to  be  fixed  on  a 
straight  rod  which  is  assumed  to  be  without  weight ;  one  ball, 
weighing  2  Ibs.,  is  fixed  at  a  distance  of  10  inches  from  the  axis 
of  motion ;  another,  which  weighs  4  Ibs.,  at  6  inches'  distance ; 
and  the  third,  which  weighs  6  Ibs.,  at  4  inches'  distance;  then 
the  distance  of  the  centre  of  gyration  from  the  axis  of  motion 
will  be  found  thus :  10  inches  squared  —100 ;  x  2  Ibs.  —  200 ; 


114  MECHANICS    OF   THE    STEAM-ENGINE. 

6  inches  squared  x  4  Ibs.  =- 144 ;  and  4  inches  squared  x  6  Ibs. 
•=  96.  The  sum  of  these  products  is  440,  which  divided  by  the 
sum  of  the  weights,  or  12  Ibs.  =  36'66,  the  square  root  of  which, 
6'05  inches,  is  the  distance  of  the  centre  of  gyration  from  the 
axis  of  motion ;  therefore,  a  single  ball  of  12  Ibs.  weight,  placed 
at  6-05  inches  from  the  axis  of  motion,  and  making  the  same 
number  of  revolutions  in  any  given  time,  would  have  the  same 
amount  of  mechanical  power  resident  in  it  as  the  three  balls  in 
their  several  places,  as  at  first  supposed. 

The  mechanical  power  which  must  be  communicated  to  a 
straight  uniform  rod  or  lever,  to  put  it  in  motion,  about  one  of 
its  extremities,  as  a  fixed  centre  or  axis,  is  the  same  as  that 
which  must  be  communicated  to  an  equal  weight  of  matter  to 
give  it  motion  in  a  straight  line,  with  '57,735  of  the  velocity 
with  which  the  extremity  of  the  lever  moves  in  its  circle.  The 
point  in  the  revolving  lever  which  moves  with  that  velocity  is 
the  centre  of  gyration. 

The  mechanical  power  which  must  be  communicated  to  a 
solid  circular  wheel  to  make  it  revolve  upon  its  axis,  is  the  same 
as  that  which  must  be  communicated  to  an  equal  weight  of 
matter  to  give  it  motion  in  a  straight  line  with  '7071  of  the 
velocity  with  which  the  periphery  of  the  wheel  moves  within 
its  circle,  and  the  point  in  the  radius  of  the  wbe«l  which  moves 
with  '7071  of  the  velocity  of  the  circumference  is  the  centre  of 
gyration.  The  weight  of  the  revolving  body,  multiplied  into 
the  height  due  to  the  velocity  with  which  the  centre  of  gyration 
moves  in  its  circle,  in  all  cases  represents  the  mechanical  power 
which  must  be  expended  upon  the  body  to  give  it  the  velocity 
of  rotation  that  it  possesses. 

THE  PENDULUM. 

The  point  from  which  the  pendulum  is  hung  is  termed  the 
centre  of  suspension.  The  effective  centre  of  the  ball  is  an 
imaginary  point  called  the  centre  of  oscillation,  and  which  is  so 
situated  that  the  distance  from  the  centre  of  suspension  to  the 
centre  of  oscillation  is  the  same  as  if  the  rod  of  the  pendulum 
were  destitute  of  weight,  and  the  whole  matter  of  the  ball  were 


LAWS    OF   THE    PENDULUM.  115 

collected  into  the  centre  of  oscillation.  The  centre  of  oscilla- 
tion is  situated  in  a  line  passing  between  the  centre  of  suspen- 
sion and  the  centre  of  gravity. 

The  number  of  vibrations  made  by  pendulums  of  different 
lengths  is  inversely  as  the  square  roots  of  their  lengths.  The 
length  of  the  pendulum  which  \vill  make  one  vibration  every 
second  is  somewhat  different  at  different  parts  of  the  earth's 
surface,  but  in  the  latitude  of  London  its  length  is  variously 
stated  at  39-1393  inches  and  39-1386  inches. 

TO   FIND    THE    HEIGHT    THROUGH    WHICH    A    BODY    WILL    FALL    IN 
THE   TIME   THAT   A   PENDULUM   MAKES   ONE   VIBRATION. 

EULE. — Multiply  the  length  of  the  pendulum  by  4'9348  and  it 
will  give  the  height. 

Example.— If  we  take  the  length  of  the  seconds  pendulum 
at  39-1386  in.,  then  39-1386  x  4-9348=193-141  in.,  which  is  the 
height  that  a  body  will  fall  by  gravity  in  a  second. 

TO   FIND   THE   LENGTH   OF   A   PENDULUM   WHICH   WILL   PEBFOBM  A 
GIVEN  NUMBEB   OF   VIBEATIONS   IN   A   MINUTE. 

RULE. — Divide  the  constant  number  375-36  by  the  number  of 
vibrations  to  be  made  per  minute,  and  the  square  of  the  quo- 
tient is  the  length  of  the  pendulum  in  inches. 

Example. — If  the  pendulum  has  to  make  60  vibrations  per 
minute,  then  375'36-=-60=6-256,  the  square  of  which  is  39-1386. 
The  length  39-1393  is  probably  still  more  nearly  the  correct 
length  of  the  seconds  pendulum  in  London. 

TO    FIND    THE    NUMBEB    OF    VTBBATION3    PEE    MINUTE     WHICH   A 
PENDULUM   OF   A   GIVEN  LENGTH   WILL  MAKE. 

RULE.— Multiply  the  square  root  of  the  length  of  the  seconds 
pendulum  ~by  the  number  of  vibrations  it  makes  per  minute, 
and  divide  the  product  by  the  square  root  of  the  length  of 
the  pendulum  whose  rate  of  vibration  has  to  be  found.  The 
quotient  is  the  number  of  vibrations  per  minute  that  the 
pendulum  will  make. 


116  MECHANICS   OF  THE   STEAM-ENGINE. 

Example.— It  the  length  of  a  pendulum  in  the  latitude  of 
London  be  28*75  inches,  what  will  be  the  number  of  vibrations 
that  it  will  make  per  minute  ? 

Here  the  square  root  of  39'1393  multiplied  by  60,  and  divided 
by  the  square  root  of  28-75=70  vibrations  per  minute. 

TO  FIND  THE  LENGTH  OF  A  PENDULUM  WHICH  SHALL  MAKE  A 
GIVEN  NUMBER  OF  VIBRATIONS  IN  A  GIVEN  TIME  IN  THE 
LATITUDE  OF  LONDON. 

EULE. — Multiply  the  square  of  the  number  of  seconds  in  the 
given  time  by  the  constant  number  39'1393,  and  divide  the 
product  by  the  square  of  the  number  of  vibrations  ;  the  quo- 
tient will  he  the  required  length  of  pendulum  in  inches. 

Example. — What  must  be  the  length  of  a  pendulum  in  order 
to  give  35  vibrations  per  minute? 

The  number  of  seconds  in  the  given  time  is  60,  hence  60 
multiplied  by  60  multiplied  by  391393  gives  140901-48,  which 
divided  by  1225  (the  square  of  35)  gives  115-021  inches,  the 
length  of  pendulum  required. 

TO   FIND   THE   NUMBER   OF   VIBRATIONS   WHICH   WILL   BE   MADE   IN 
A   GIVEN  TIME   BY   A   PENDULUM   OF   A   GIVEN   LENGTH. 

EULE. — Multiply  the  square  of  the  number  of  seconds  in  the 
given  time  by  the  constant  number  39-1393,  divide  the  prod- 
uct by  the  given  length  of  the  pendulum  in  inches,  and  the 
square  root  of  the  quotient  will  be  the  number  of  vibrations 
in  the  given  time. 
Example. — The  length  of  a  pendulum  being  64  inches,  what 

number  of  vibrations  will  it  make  in  60  seconds? 

In  this  case  the  square  of  60  multiplied  by  39*1393  gives 

140901-48,  which  being  divided  by  64  gives  2201-5856,  the  square 

root  of  which  46'09  is  the  number  of  vibrations  required. 

THE  GOVERNOR. 

The  governor  is  a  centrifugal  pendulum ;  and  its  proportions 
may  be  fixed  by  the  same  rules  which  are  employed  to  deter- 


REVOLVING  PENDULUM  OB  GOVERNOR.      117 

mine  the  rates  of  vibration  of  pendulums.  If  we  suppose  a  pen- 
dulum, in  the  act  of  vibration,  to  be  at  the  same  time  pushed 
sideways  by  a  suitable  force,  it  will  nevertheless  perform  its 
vibration  in  the  same  period  of  time ;  and  if  during  its  return  it 
be  again  pushed  sideways  in  the  opposite  direction,  it  will, 
during  this  double  vibration,  have  pursued  a  curvilinear  course, 
which,  if  the  deflection  be  sufficient,  will  be  a  circle.  A  pendu- 
lum, therefore,  of  the  same  vertical  height  as  the  cone  described 
by  the  arms  of  a  governor,  will  perform  a  double  vibration  in  the 
same  time  as  the  governor  performs  one  revolution.  The  rules, 
however,  according  to  which  governors  are  usually  proportioned 
are  as  follow  : — 

TO  DETERMINE  THE  PEOPEE  HEIGHT  OF  THE  POINT  OF  SUSPEN- 
SION OF  THE  BALLS  OF  A  GOVEBNOB,  ABOVE  THE  PLANE  IN 
WHICH  THEY  BEVOLVE  WHEN  MOVING  WITH  MEAN  VELOCITY. 

RULE. — Divide  the  number  35,225  T>y  the  square  of  the  main 
number  of  revolutions  which  the  governor  makes  per  minute. 
The  quotient  is  the  proper  vertical  height  in  inches  of  the 
point  of  suspension  of  the  tails  above  the  plane  in  which 
they  revolve,  when  moving  with  mean  velocity. 

Example. — "What  is  the  proper  vertical  height  of  the  point 
of  suspension  above  the  plane  of  revolution  in  the  case  of  a  gov- 
ernor making  30  revolutions  per  minute? 

Here  35225-^900  (the  square  of  30)  =  39-139,  which  is  the 
same  height  as  that  of  the  seconds  pendulum. 

If  we  have  already  the  vertical  height,  and  wish  to  know  the 
proper  tune  of  revolution,  we  must  proceed  as  follows : — 

TO    DETEBMINE    THE    PEOPEE    TIME    OF    BEVOLTTTION    OF  A  GOV- 
EENOB  OF  WHICH  THE  VEBTICAL  HEIGHT  18  KNOWN. 

RULE. — Multiply  the  square  root  of  the  height  'by  the  constant 
fraction  0-31986,  and  the  product  will  be  the  proper  time  of 
revolution  in  seconds. 

Example.— In  what  time  should  a  governor  be  made  to  re- 
volve upon  its  axis  when  the  vertical  height  of  the  cone  in  which 


118 


MECHANICS    OF   THE   STEAM-ENGINE. 


the  arms  are  required  to  revolve  when  in  their  mean  position  is 
39-1393  inches  ?     Here  6'256  x  0-31986=2  seconds. 

FRICTION. 

When  two  hodies  are  rubbed  together  they  generate  heat, 
and  consume  thereby  an  amount  of  power  which  is  the  mechan- 
ical equivalent  of  the  heat  produced.  Clean  and  smooth  iron 
drawn  over  clean  and  smooth  iron  without  the  interposition  of 
a  film  of  oil,  or  other  lubricating  material,  requires  about  one- 
tenth  of  the  force  to  move  it  that  is  employed  to  force  the  sur- 
faces together.  In  other  words,  a  piece  of  iron  10  Ibs.  in  weight 
would  require  a  weight  of  1  Ib.  acting  on  a  string  passing  over  a 
pulley  to  draw  the  10  Ib.  weight  along  an  iron  table.  But  if  the 
surfaces  are  amply  lubricated,  the  friction  will  only  be  from 
^th  to  sLth  of  the  weight.  The  friction  of  cast-iron  surfaces  in 
sandy  water  is  about  one-third  of  the  weight.  The  extent  of  the 
rubbing  surface  does  not  affect  the  amount  of  the  friction. 

The  experiments  of  General  Morin  on  the  friction  of  various 
bodies  without  an  interposed  film  of  lubricating  liquid,  but  with 
the  surfaces  wiped  clean  by  a  greasy  cloth  have  been  summarised 
by  Mr.  Eankine  in  the  following  table : — 

GENERAL  MORIN's  EXPERIMENTS   ON  FRICTION. 


No. 

SURFACES. 

Angle  of 
repose. 

Friction  In 
terms  of 
the  weight. 

1 

Wood  on  wood,  dry  

14°  to  26}° 

•25  to  5 

2 

11}°  to  2' 

•2  to  -04 

8 

Metals  on  oak,  dry  

26}°  to  81° 

•5  to  -6 

4 

"    wet  

13}°  to  14}° 

•24  to  -26 

5 

"    soapy  

ii}° 

•2 

6 

Metals  on  elm,  dry  

11}°  to  14° 

•2  to  -25 

7 

Hemp  on  oak,  dry  

28° 

•53 

8 

"    wet  

18}° 

•88 

9 

Leather  on  oak  

15°  to  19}° 

•27  to  -88 

10 

Leather  on  metals,  dry  

29  }' 

•56 

11 

"          wet  

20' 

•36 

12 

"              "           greasy  .  .  . 

18° 

•23 

18 

oily  

8}° 

•15 

14 

Metals  on  metals,  dry  

8}°  to  11}° 

•15  to  -2 

15 

"               "         wet  

16}° 

•8 

16 

IT 
18 

Smooth  surfaces,  occasionally  greased  .  .  . 
"        continually  greased... 
"       best  results  

4°  to  4}° 
8° 
1}°  to  2° 

•07  to  -03 
•05 
•08  to  -086 

19 

Bronze  on  lignum  vita;,  constantly  wet  . 

3°? 

•05? 

LAWS   OF  FRICTION.  119 

The  '  Angle  of  repose,'  given  in  the  first  column,  is  the  angle 
which  a  flat  surface  will  make  with  the  horizon  when  a  weight 
placed  upon  it  just  ceases  to  move  by  gravity.  The  column  of 
'  Friction  in  terms  of  the  weight  '  means  the  proportion  of  the 
weight  which  must  be  employed  to  draw  the  body  by  a  string  in 
order  to  overcome  its  friction  ;  and  the  proportional  weight  is 
sometimes  called  the  Co-efficient  of  Friction. 

In  a  paper,  of  which  an  abstract  has  appeared  in  the  Comptes 
Bendus  of  the  French  Academy  of  Sciences  for  the  26th  of  April, 
1858,  M.  H.  Bochet  describes  a  series  of  experiments  which  have 
led  him  to  the  conclusion,  that  the  friction  between  a  pair  of 
surfaces  of  iron  is  not,  as  it  has  hitherto  been  believed,  absolutely 
independent  of  the  velocity  of  sliding,  but  that  it  diminishes 
slowly  as  that  velocity  increases,  according  to  a  law  expressed 
by  the  following  formula.  Let 

R  denote  the  friction  ; 

Q,  the  pressure  ; 

t>,  the  velocity  of  sliding,  in  metres  per  second  =  velocity  in 
feet  per  second  x  0-3048  ; 

/,  a,  7,  constant  co-efficients  ;  then 
B 


The  following  are  the  values  of  the  co-efficients  deduced  by 
M.  Bochet  from  his  experiments,  for  iron  surfaces  of  wheels  and 
skids  rubbing  longitudinally  on  iron  rails  :  — 

/,  for  dry  surfaces,  0-3,  0-25,  0'2  ;  for  damp  surfaces,  0.14. 

a,  for  wheels  sliding  on  rails,  0'03  ;  for  skids  sliding  on  rails, 
0-07. 

y,  not  yet  determined,  but  treated  meanwhile  as  inapprecia- 
bly small. 

The  friction  of  a  bearing  or  machine  per  revolution,  is  nearly 
the  same  at  all  velocities,  the  pressure  being  supposed  to  be 
uniform  ;  but  as  every  revolution  absorbs  a  definite  quantity  of 
power,  and  generates  a  corresponding  quantity  of  heat,  it  will 
be  necessary  to  enlarge  the  rubbing  surfaces  at  high  velocities, 
both  to  prevent  the  wear  from  being  inconveniently  rapid,  and 
also  to  enable  the  bearing  to  present  a  larger  cooling  surface  to 


120  MECHANICS   OP  THE   STEAM-ENGINE. 

the  atmosphere,  so  as  to  disperse  the  increments  of  heat  which 
in  the  case  of  high  velocities  it  will  rapidly  receive.  With  the 
same  object  the  lubrication  should  be  ample.  The  oil  should 
overflow  the  bearing,  in  the  same  manner  as  the  oil  in  a  carcel 
or  moderator  lamp  overflows  the  wick  to  prevent  carbonisation ; 
and,  to  prevent  waste,  the  oil  should  be  returned  by  an  oil 
pump  so  as  to  maintain  a  circulation  that  will  both  cool  and  lu- 
bricate the  rubbing  parts. 

It  was  found  by  General  Morin  in  his  experiments,  that  the 
'  Friction  of  Eest '  was  considerably  more  than  the  '  Friction 
of  Motion,'  or,  in  other  words,  that  it  took  a  greater  force  to 
move  a  rubbing  body  from  a  state  of  rest  than  it  afterwards 
took  to  continue  the  motion,  some  of  the  softer  bodies  being  in 
fact  slightly  indented  with  the  weight.  But  in  determining  the 
friction  of  machinery,  it  is  the  friction  of  motion  alone  that  has 
to  be  considered,  so  that  the  other  need  not  be  here  taken  into 
account. 

In  the  case  of  rubbing  surfaces  which  are  amply  lubricated, 
the  amount  of  the  friction  depends  more  on  the  nature  of  the 
lubricant  than  upon  the  material  of  which  the  rubbing  bodies 
are  composed;  and  hard  lubricants,  such  as  tallow,  are  more 
suited  for  heavy  pressures ;  and  thin  lubricants,  such  as  almond 
oil,  are  best  suited  for  mechanisms  moving  with  considerable 
velocity,  but  on  which  the  strain  is  small.  If  too  heavy  a  press- 
ure be  applied  to  a  bearing,  the  oil  will  be  forced  out  and  the 
surfaces  will  heat ;  and  this  will  be  liable  to  take  place  when 
the  pressure  on  the  bearing  is  much  more  than  800  Ibs.  per 
square  inch  on  the  longitudinal  section  of  the  bearing,  though 
in  practice  the  pressure  is  sometimes  half  as  much  again,  or 
about  1,200  Ibs.  per  square  inch  in  the  longitudinal  section  of 
the  bearing,  but  such  bearings  will  be  liable  to  heat.  Thus  in  a 
marine  engine  with  a  cylinder  of  74^  inches  diameter,  the  crank 
pin  is  9£  inches  diameter,  and  the  length  of  the  bearing  is  10 
inches,  which  makes  the  area  of  the  longitudinal  section  of  the 
bearing  95  square  inches.  The  area  of  the  cylinder  is  4,359 
square  inches,  and  if  we  take  the  pressure  upon  the  piston — 
including  steam  and  vacuum — at  25  Ibs.  per  square  inch,  we 


PRESSURE   FOR  A   GIVEN  VELOCITY.  121 

shall  have  a  total  pressure  on  the  piston  of  108,975  Ibs.,  and, 
consequently,  this  amount  of  pressure  on  the  crank  pin  bearing. 
Now  108,975  Ibs.,  the  total  pressure,  divided  by  95  square 
inches,  the  total  surface,  gives  1,147  Ibs.  for  each  square  inch 
of  the  parallelogram  which  forms  the  longitudinal  section  of  the 
bearing.  In  the  engines  of  Messrs.  Maudslay,  Messrs.  Seaward, 
and  most  of  the  London  engineers,  the  pressure  per  square  inch 
put  upon  the  crank  pin  is  less.  Thus  in  their  120-horse  engines, 
the  diameter  of  the  cylinder  is  57|  inches,  giving  an  area  of 
2,597  square  inches,  which  multiplied  by  a  pressure  of  25  Ibs. 
per  square  inch,  gives  64,925  Ibs.  as  the  total  pressure  upon  the 
piston.  The  crank  pin  is  8  inches  diameter,  and  the  bearing  is 
8£  inches  long,  giving  68  square  inches  as  the  area  of  the  longi- 
tudinal section ;  and  64,925  Ibs.,  the  total  pressure,  divided  by 
68  square  inches,  the  total 'area,  gives  a  pressure  of  954'771bs. 
per  square  inch  of  section.  This  is  still  in  excess  of  the  800 
Ibs.  per  square  inch  to  which  it  is  expedient  to  limit,  the  press- 
ure. But  the  assumed  pressure  on  the  piston  is  rather  large 
in  the  case  of  these  engines,  and  the  actual  pressures  will  be 
found  to  agree  pretty  well  with  the  limit  of  800  Ibs.  on  each 
square  inch  of 'the  longitudinal  section  of  bearings  which  it  is 
proper  to  fix  as  a  general  rule  in  the  case  of  engines  moving 
slowly.  In  the  case  of  fast-moving  engines,  however,  the  sur- 
face should  be  greater.  The  proportion  in  which  the  surface 
should  vary  with  the  speed  is  pretty  accurately  expressed  by  the 
following  rule : — 

TO  FIND  THE  PRESSURE  PER  SQUARE  INCH  THAT  MAT  BE  PUT 
UPON  A  BEARING  MOVING  WITH  ANT  GIVEN  VELOOITT. 

RULE. — To  the  constant  number  50  add  the  velocity  of  the  'bear- 
ing in  feet  per  minute,  and  reserve  the  sum  for  a  divisor. 
Divide  the  constant  number  70,000  ty  the  divisor  found  as 
above.  The  quotient  will  le  the  number  of  pounds  per  square 
inch  that  may  "be  put  upon  the  bearing. 

Example  1. — An  engine  with  a  cylinder  74£  inches  diameter, 
has  a  crank  pin  10  inches  diameter.    At  220  feet  of  the  piston 
6 


122  MECHANICS   OF   THE   STEAM-ENGINE. 

per  minute,  and  with  a  stroke  of  7J  feet,  the  number  of  revolu- 
tions per  minute  will  be  about  15  ;  and  as  the  circumference  of 
the  crank  pin  will  be  about  30  inches  or  2£  feet,  the  surface  of 
the  bearing  will  travel  15  times  2|,  or  37£  feet  per  minute. 
Adding  to  this  the  constant  number  50,  we  have  87£,  and  70,000 
divided  by  87£  =  800,  which,  at  this  speed,  is  the  proper  press- 
ure to  put  on  each  square  inch  of  the  longitudinal  section  of 
the  bearing.  If  it  is  found  on  trial  that  this  pressure  is  ex- 
ceeded, the  length  or  diameter  of  the  pin  must  be  increased 
or  both. 

Example  2. — An  engine  with  a  cylinder  42  inches  diameter, 
has  a  crank  pin  8J  inches  diameter,  the  circumference  of  which 
is  26'7  inches  or  2-225  feet.  When  the  engine  makes  54-8  revo- 
lutions per  minute,  the  surface  of  the  crank  pin  will  move  with 
a  speed  of  54-8  times  2'225  feet  per  minute,  or  121-8  feet 
per  minute.  Now  50  +  121-8  =  171 '8,  and  70,000  divided  by 
171'8=407'3,  which,  at  this  speed  of  revolution,  is  the  proper 
load  to  place  upon  each  square  inch  of  section  in  the  line  of  the 
axis.  The  pressure  of  steam  and  vacuum  in  this  engine  was  40 
Ibs.  per  square  inch ;  and  as  the  area  of  a  piston  42  inches  diam- 
eter is  1385*4  square  inches,  the  pressure  urging  the  piston  will 
be  40  times  1385-4  or  55,416  Ibs.  Now  55,416  divided  by  407'3 
is  136,  which  must  be  the  number  of  square  inches  in  the  longi- 
tudinal section  of  the  bearing  in  order  that  there  may  not  be 
more  than  407'3  Ibs.  on  each  square  inch.  To  obtain  this  area, 
the  bearing  must  be  16  inches  long,  since  8£  inches  multiplied 
by  16  inches  is  136  square  inches.  At  60  revolutions,  the  speed 
of  the  bearing  surface  per  minute  is  60  tunes  2-225  feet  or  133-5 
feet.  Now  50  + 133-5=183-5,  and  70,000  divided  by  183-5=377'4, 
which  is  the  proper  load  in  Ibs.  for  each  square  inch  in  the  lon- 
gitudinal section  of  the  bearing.  At  70  revolutions  the  speed 
of  the  bearings  is  70  times  2-225  feet,  or  155-75  feet  per  minute. 
Now  50  +  155-75=205-75,  and  70,000  divided  by  205-75=340-2, 
which  is  the  proper  load  in  pounds  to  put  upon  each  square 
inch  of  the  longitudinal  section  of  the  bearing  at  this  speed  of 
rotation. 


PRESSURE    FOR   A   GIVEN  VELOCITY.  123 

TO  FIND  THE  PROPER  TELOCITY  FOR  THE  SURFACE  OF  A  BEAR- 
ING WHEN'  THE  PRESSURE  PER  SQUARE  INCH  ON  ITS  LONGITU- 
DINAL SECTION  IS  GIVEN. 

RULE. — Divide  the  constant  number  70,000  ly  the  pressure  per 
square  inch  on  the  longitudinal  section  of  the  tearing,  from 
the  quotient  subtract  the  constant  number  50.  The  remain- 
der is  the  proper  velocity  of  the  surface  of  the  bearing  in 
feet  per  second. 

Example  1. — "What  is  the  proper  velocity  of  the  surface  of  a 
bearing  which  has  the  pressure  of  800  Ibs.  on  each  square  inch 
of  its  longitudinal  section  ?  Here  70,000  divided  by  800=87'5 ; 
from  which  if  we  take  50  there  will  remain  37*5,  which  is  the 
proper  velocity  of  the  bearing  in  feet  per  second. 

If  we  take  a  hypothetical  pressure  of  1,400  Ibs.  per  square 
inch  of  section,  we  get  70,000  divided  by  1,400  =  50,  and 
50—50=0;  so  that  with  such  a  pressure  there  should  be  no 
velocity.  Even  in  cases,  however,  in  which  there  is  very  little 
motion,  such  as  in  the  top  eyes  of  the  side  rods  of  marine 
engines,  it  is  not  advisable  to  have  so  great  a  pressure  upon 
the  bearing  as  1,400  or  even  1,200  Ibs.  per  square  inch  of 
section. 

Example  2. — What  is  the  proper  velocity  of  the  bearing  of 
an  engine  which  has  a  pressure  upon  it  of  407'3  Ibs.  per  square 
inch  of  section?  Here  70,000  divided  by  407*3=1 71 -8,  which 
diminished  by  50  is  121*8,  which  is  the  proper  speed  of  the  sur- 
face of  the  bearing  with  this  pressure  per  square  inch  of  sec- 
tion. If  the  diameter  of  the  bearing  be  8£  inches,  its  circum- 
ference will  be  2-225  feet,  and  121-8  divided  by  2-225=54-8  rev- 
olutions, which  will  be  the  speed  of  the  engine  with  these  data. 
These  proportions  allow  a  good  margin,  which  may  often  be 
availed  of  in  practice,  either  in  driving  the  engine  faster  than 
is  here  indicated,  or  in  putting  more  pressure  upon  the  bearing. 
But  to  obviate  inconvenient  heating  and  wear,  it  will  be  found 
preferable  to  adhere,  as  nearly  as  practicable,  to  the  proportion 
of  surface  which  these  rules  prescribe. 


124  MECHANICS    OP   THE    STEAM-EXGINE. 


STRENGTH  OF  MATERIALS. 

The  various  kinds  of  strain  to  which  materials  are  exposed 
in  machines  and  structures  may  be  all  resolved  into  strains  of 
extension  and  strains  of  compression ;  and  in  investigating  the 
strength  of  materials  there  are  three  fixed  points,  varying  in 
every  material,  to  which  it  is  necessary  to  pay  special  regard — 
the  ultimate  or  breaking  strength,  the  elastic  or  proof  strength, 
and  the  safe  or  working  strength.  The  tensile  or  breaking 
strength  of  wrought-iron,  is  about  60,000  per  square  inch  of 
section,  whereas  the  crushing  strength  of  wrought-iron  is  about 
27,000  per  square  inch  of  section.  In  steam-engines  where  the 
parts  are  alternately  compressed  and  extended,  it  is  not  proper 
to  load  the  wrought-iron  with  more  than  4,000  Ibs.  per  square 
inch  of  section ;  or  the  cast-iron  with  more  than  2,000  Ibs.  per 
square  inch  of  section.  But  in  boilers  where  the  strain  is  con- 
stantly in  one  direction,  the  load  of  4,000  Ibs.  per  square  inch 
of  section  may  be  somewhat  exceeded.  The  elastic  strength  is 
the  strength  exhibited  by  any  material  without  being  perma- 
nently altered  in  form,  or  crippled ;  for  as  a  piece  of  iron  is 
finally  broken  by  being  bent  backward  and  forward,  so  by  ap- 
plying undue  strains  to  any  material,  it  will  be  finally  broken 
with  a  much  less  strain  than  would  suffice  to  break  it  at  once. 
The  elastic  tensile  strength  of  wrought-iron  is  between  one-third 
and  one-fourth  of  its  ultimate  tensile  strength,  and  to  this  point 
the  material  might  be  proved  without  injury.  But  in  proving 
boilers,  and  many  other  objects,  it  is  not  usual  to  make  the 
proving  pressure  more  than  twice  or  three  times  the  working 
pressure,  such  proof  it  is  considered  involving  no  risk  of  strain- 
ing the  material  while  it  is  adequate  to  the  detection  of  acci- 
dental flaws  if  such  exist.  The  following  table  exhibits  the  te- 
nacity or  tensible  strength,  and  the  resistance  to  compression  or 
crushing  strength  of  various  materials : — 


STRENGTH    OF   MATERIALS. 


125 


TENSILE  AND  CRUSHING  STRENGTHS  OF  VARIOUS  MATERIALS  PER  SQUARE  INCH 
OF  SECTION. 


MATERIAL. 

Tensile  strength 
in   Ibs.   per  square 
inch  of  section. 

Crashing  strength 
in    Ibs.    per    square 
inch  of  section. 

METALS. 
Wrought-iron  bars  

60,000 
62.000       1 
64,000 
j     70,000  to  f 
1  100,000      J 
36,500 
25,764 
(  100,000  to 
)  180,000 
18,000 
36,000 
50,000 
19,000 
30,000 
88,000 
60,000 
40,997 
20,490 
4,736 
8,137 
7,000 
1,062 
8,000 

17,000 
12,000 
15,000 
20,000 
18,000 
j  10,000  to 
1  14,000 
20,000 
28,000 
12,000 
16,000 
i     8,000  to  1 
16,000       f 
10,000  to  } 
19,000       f 
9,800 
15,000 

j  10,666  to 

|  12,000 

27,000  to  37,000 

varies    as    cube 
of     thickness 
nearly. 

100,000 
130,000 

j.    260,000 
10,000 

9,000t 
9,300 
6,400 
10,800 
10,800 
6,875  to  ) 
6,200       f 
7,800 

9,900 

8,200 
10,000 

12,000 

5,500  to» 
11,000       f 
4,000  to  ( 
5,000       f 

4,000  to) 
6,000       f 
660  to  i 
800       j 
1,100 
1,700 

Wrought-iron   plates  

Wrought-iron  hoops  (best  best)  

Wrought-iron  wire*  

Cast-Iron  (average)  

Cast-iron  (toughened)  

Steel                          

Cast  brass  

Brass  wire  

Cast  copper  

Silver  (cast)        

Gold  

Tin  (cast)  

Bismuth  (cast)  

Zinc  

Antimony  

Lead  (sheet)  

WOODS. 

Ash  

Beech  

Birch  

Box  

Elm  

Fir  (red  pine)  

Hornbeam  

Lance-wood  

Lignum  Yltse  

Locust  

Mahogany  

Oak  

Pear  ... 

Teak  

STONES. 

Granite  

Limestone  

Slate  

Sandstone  

Brick  (weak)  

Brick  (strong)  

Brick  (fire)  

Glass  

9,600 
60 

Mortar  

*  Mr.  Pole  found  the  German  steel  wire  used  for  pianoforte*  to  bear  a>  much  u  268,800  Iba,  per 
iqnare  Inch, 
t  Those  value*  are  for  dry  wood.    In  wet  wood  the  crushing  strength  U  only  half  a*  great. 

126 


MECHANICS    OF   THE    STEAM-ENGINE. 


It  will  be  remarked  that  there  are  very  large  variations  in 
the  amount  of  the  strength  recorded  in  this  table ;  and  there  are 
so  many  varieties  in  the  quality  of  the  materials  experimented 
upon  that  it  is  hopeless  to  expect  any  absolute  agreement  in  the 
results  of  different  experiments.  It  will  be  useful,  under  these 
circumstances,  to  set  down  the  main  results  arrived  at  by  a  few 
of  the  principal  experimentalists,  leaving  the  reader  to  select 
such  value  as  he  may  consider  most  nearly  agrees  with  the  cir- 
cumstances with  which  he  has  to  deal,  The  following  are  the 
strengths  of  various  metals  ascertained  by  Mr.  George  Eennie, 
in  1817:— 


TENSILE  STRENGTHS  OF  METALS  BY  RENNIE. 


KIND   OF  METAL. 

Tearing    weight 
in  Ibs.  of  a  bar 
one  inch 
square. 

Length  of  bar 
one  inch  square 
in  feet  that 
would  break  by 
its  own  weight. 

Cast  steel  

134,256 

39,455 

Swedish  malleable  iron  

72,064 

19,740 

English  malleable  iron  

55,872 

16  938 

Cast-iron  

19,096 

6  110 

Cast  copper  

19,072 

5,003 

Yellow  brass  

17,958 

5,180 

Case  tin  

4,736 

1,496 

Cast  lead  

1,824 

348 

Professor  Leslie,  in  his  Natural  Philosophy,  thus  explains 
the  law  of  the  extension  of  iron  by  weights  :— 

4  A  bar  of  soft  iron  will  stretch  uniformly  by  continuing  to 
append  to  it  equal  weights  till  it  can  be  loaded  with  half  as 
much  as  it  can  bear ;  beyond  that  limit,  however,  its  extension 
will  become  doubled  by  each  addition  of  the  eighth  part  of  the 
disruptive  force.  Suppose  the  bar  to  be  an  inch  square  and 
1,000  inches  in  length;  36,000  Ibs.  will  draw  it  out  1  inch,  but 
45,000  will  stretch  it  2  inches;  54,000  Ibs.  4  inches;  63,000 
8  inches;  and  72,000  16  inches,  where  it  would  finally  break.' 
This  popular  explanation  of  the  law  agrees  pretty  nearly 


TENSILE   AND   CRUSHING   STRENGTHS. 


127 


with    the  subsequent  deductions   of   Hodgkinson    and    other 
enquirers. 

The  cohesive  strength  of  woods  varies  still  more  than  that 
of  metals  in  different  specimens,  and  varies  even  in  different 
parts  of  the  same  tree.  Thus  in  Barlow's  experiments  he  found 
the  cohesive  strength  of  fir  to  vary  from  11,000  to  13,448  Ibs. 
per  square  inch  of  section ;  of  ash  from  15,784  to  17,850  ;  oak 
from  8,889  to  12,008;  pear  from  8,834  to  11,537,  and  other 
woods  in  the  same  proportions.  The  following  fair  average 
values  may  he  adopted : — 


TENSILE   STRENGTHS   OP   WOODS   BY   BARLOW. 


KIND   OF  WOOD. 

Tearing 
weight  in  Ibs. 
of  a  rod  one 
inch  square. 

Length  in  feet 
of  a  rod  one 
inch  square  that 
would  break  by 
its  own  weight. 

Teak  

12,915 

36,049 

Oak  

11,880 

32,900 

Sycamore  

9,630 

35,800 

Beech     

12,225 

38,940 

Ash  

14,130 

42,080 

Elm  

9.720 

39,050 

9,540 

40,500 

Christiana  Deal  

12,846 

55,500 

Larch  

12,240 

42,160 

The  crushing  strength  of  wood,  as  of  most  other  materials, 
is  very  different  from  its  tensile  strength,  and  is  greatly  affected 
by  its  dryness.  The  following  table  exhibits  the  results  of  the 
experiments  made  by  Mr.  Hodgkinson,  to  ascertain  the  crush- 
ing strengths  of  different  woods  per  square  inch  of  section. 
The  specimens  crushed  were  short  cylinders,  1  inch  diameter 
and  2  inches  long,  flat  at  the  ends.  The  results  given  in  the 
first  column  are  those  obtained  when  the  wood  was  moderately 
dry.  Those  in  the  second  column  were  obtained  from  similar 
specimens  which  had  been  kept  two  months  longer  in  a  warm 
place : — 


128 


MECHANICS    OF   THE    STEAM-ENGINE. 


STEEXGTHS   OF   WOODS   BY   IIODGKINSON. 


KIND   OF   WOOD. 

Crushing  strength  per 
square  inch  of  section. 

Alder  

6831     to      6960 

Ash    

8683      "      9363 

Bay  .  . 

7518      "      7518 

Beech  

7733      "      7363 

English  Birch  

3297      "      6402 

5674     "      5863 

Red  Deal  

5748      "      6586 

White  Deal  

6781      "      7293 

Elder  

7451      "      9973 

Elm                

"    10331 

Fir  (Spruce)  

6499      "      6819 

Mahogany  

8198      "      8198 

Oak   (Quebec)  

4231       '      5982 

Oak  (English)  

6484       '    10058 

Pine  (Pitch)  

6790       '      6790 

Pine  (Red)       

5395       '      7518 

Poplar                       

3107       '      5124 

Plum  (Dry)  

8241       '    10493 

Teak  

'    12101 

Walnut  

6063      "      7227 

Willow  

2898      "      6128 

The  crushing  strength  of  cast-iron  is  98,922  Ihs.,  or,  say 
100,000  per  square  inch  of  section. 

The  strength  of  wooden  columns  of  different  lengths  and 
diameters  to  sustain  weights  has  not  been  conclusively  deter- 
mined, and  the  longer  a  column  is  the  weaker  it  is.  But,  how- 
ever short  it  may  be,  the  load  upon  it  should  not  be  above  one- 
third  of  the  crushing  load,  as  given  above. 

LAW  OF  THE  STRENGTH  OF  PILLARS. 

The  theory  of  the  strength  of  pillars  propounded  by  Euler  is 
that  the  strength  varies  as  the  fourth  power  of  the  diameter 
divided  by  the  square  of  the  length ;  and  the  recent  investiga- 
tions of  Hodgkinson  and  others  show  that  this  doctrine  is  nearly 
correct.  Thus,  in  the  case  of  hollow  cylindrical  columns  of  cast- 
iron,  it  is  found  experimentally  that  the  3-55th  power  of  the  in- 
ternal diameter  subtracted  from  the  3'55th  power  of  the  external 


LAW   OF   STRENGTH   OF   PILLARS.  129 

diameter,  and  divided  by  the  l'7th  power  of  the  length,  will 
give  the  strength  very  nearly.  In  the  case  of  hollow  cylindrical 
columns  of  malleable  iron,  it  is  found  that  the  3 '5  9th  power  of 
the  internal  diameter,  subtracted  from  the  3'59th  power  of  the 
external  diameter,  and  divided  by  the  square  of  the  length,  will 
represent  the  strength  ;  but  this  rule  only  holds  when  the  load 
does  not  exceed  8  or  9  tons  per  square  inch  of  section.  The 
power  of  plates  to  resist  compression  varies  as  the  cube  or  more 
nearly  as  the  2'878th  power  of  their  thickness.  But  this  law  only 
holds  so  long  as  the  pressure  applied  does  not  exceed  9  to  12 
tons  per  square  inch  of  section.  If  the  load  is  made  greater 
than  this,  the  metal  is  crushed  and  gives  way.  It  has  been  found 
experimentally  that  in  malleable  iron  tubes  of  the  respective 
thicknesses  of  '525,  '272  and  '124  inches,  the  resistances  to  com- 
pression per  square  inch  of  section  are  19'17, 14-47,  and  7'47tons 
respectively.  Moreover,  in  wrought-iron  tubes  1£  inches  diam- 
eter and  £th  of  an  inch  thick,  the  crushing  strength  is  only  6 '55 
tons  per  square  inch  of  section,  while  in  tubes  of  nearly  the  same 
length  and  thickness,  but  about  6  inches  diameter,  the  crushing 
strength  is  16  tons  per  square  inch  of  section.  The  strength  of 
a  pillar  fixed  at  both  ends  is  twice  as  great  as  if  it  were  rounded 
at  both  ends.  The  crushing  strength  of  a  single  square  cell  or 
tube  of  wrought-iron  of  large  size,  with  angle-irons  at  the  cor- 
ners, of  the  construction  adopted  in  tubular  bridges,  is  when  the 
thickness  of  the  plate  is  not  less  than  one-thirtieth  of  the  di- 
ameter of  the  cell,  about  27,000  Ibs.  per  square  inch  of  section ; 
and  where  a  number  of  such  cells  are  grouped  together  so  as  to 
prevent  deflection,  the  crushing  strength  rises  to  nearly  36,000  Ibs. 
per  square  inch  of  section,  which  is  also  the  crushing  strength 
of  short  wrought-iron  struts.  The  length  of  independent  pillars 
should  not  be  more  than  25  tunes  the  diameter. 

The  weight  in  Ibs.  which  a  square  post  of  oak  of  any  length 
will  with  safety  sustain  may  be  determined  as  follows : — 

TO  DETERMINE  THE  PEOPEE  LOAD  FOE  OAK  POSTS. 

RULE. — To  4  times  the  square  of  the  breadth  in  inches  add 
half  the  square  of  the  length  in  feet,  and  reserve  the  sum  for 
6* 


130 


MECHANICS   OF   THE   STEAM-ENGINE. 


a  divisor.  Multiply  the  cube  of  the  'breadth  in  inches  ~by 
3,960  times  the  length  in  feet,  and  divide  the  product  by  the 
divisor  found  as  above.  The  quotient  is  the  weight  in  Ibs. 
which  the  oak  post  or  pillar  will  with  safety  sustain. 

Example. — "What  weight  will  a  column  of  oak  6  inches  square 
and  12  feet  long  sustain  with  safety? 

Here  the  breadth  of  the  post  is  6  inches,  the  square  of  which 
is  36 ;  and  4  times  36  is  144.  The  length  heing  12  feet,  the 
square  of  the  length  is  144,  half  of  which  is  72 ;  and  72  added  to 
144  gives  216  for  the  divisor.  The  breadth  being  6  inches,  the 
cube  of  the  breadth  is  216,  and  the  length  being  12  feet,  we  get 
12  times  3,960  which  is  47,520.  Then  216  times  47,520  is  10,- 
264,320,  which  divided  by  216  gives  47,520,  which  is  the  weight 
in  Ibs.  that  the  post  will  with  safety  sustain. 

The  following  table  is  computed  from  the  rule  given  above : — 


SCANTLINGS  OF  SQUARE  POSTS  OF  OAK. 

With  the  weights  they  will  support  and  the  extent  of  surface  of  flooring 
they  will  safely  sustain,  allowing  1  cwt.,  If  cwt.,  or  2  cwts.  to  the 
superficial  foot  of  flooring,  and  calculated  for  a  height  of  10  feet. 

NOTE. — These  Scantlings  may  "be  safely  used  -up  to  ~i2feet  in  height;  'but  above 
that  a  little  extra  thickness  should  be  allowed. 


Scantlings. 

Weight 

Extent  of  surface  of  flooring  supported. 

1  cwt.  per  foot. 

l£  cwt.  per  foot. 

2  cwt  per  foot. 

Inches. 

Tons.    Cwts. 

Square  feet. 

Square  feet. 

Square  feet. 

3x3 

5         10 

110 

82^ 

55 

4x4 

9         18 

198 

148£ 

99 

5x5 

14         14 

294 

220£ 

147 

6x6 

19         12 

892 

294 

196 

7x7 

24         12 

492 

369 

246 

8x8 

29         10 

590 

442£ 

295 

9x9 

34          8 

688 

516 

344 

10x10 

39          4 

784 

588 

392 

11x11 

44          0 

880 

660 

440 

12x12 

48         16 

976 

732 

488 

13x13 

53         10 

1070 

802^ 

535 

14x14 

58          4 

1164 

873 

582 

15x15 

62         16 

1256 

942 

628 

LAW   OP   STRENGTH   OF   PILLARS.  131 

Similar  calculations  of  the  dimensions  and  loads  proper  for 
rectangular  columns  of  other  woods  may  be  determined  hy  a 
reference  to  their  relative  crushing  strengths  given  in  page  128. 

The  formula  given  by  Mr.  Hodgkinson  for  determining  the 
breaking  -weight  of  square  oak  posts  where  the  length  exceeds 
30  times  the  thickness  is 

W=2452— . 
2" 

where  W  is  the  breaking  weight  in  Ibs. ;  d  the  side  of  the  square 
base  in  inches ;  and  I  the  length  of  the  post  in  feet. 


TO    DETEBMDTE    THE    PEOPEB    LOAD    TO    BE    PLACED  UPON  SOLID 
PILLABS  OF  CAST-IKON. 

The  load  which  may  be  safely  placed  upon  round  posts,  or 
solid  pillars  of  cast-iron,  may  be  ascertained  by  the  following 
rule : — 

EULE. — To  4  times  the  square  of  the  diameter  of  the  solid 
pillar  in  inches,  add  0*18  times  the  square  of  the  length,  of  the 
pillar  in  feet,  and  reserve  the  sum  for  a  divisor.  Multiply 
the  fourth  power  of  diameter  of  the  pillar  in  inches  Try  the 
constant  number  9562  and  divide  the  product  oy  the  divisor 
found  as  above.  The  quotient  is  the  weight  in  Ibs.  which  the 
solid  cylinder  or  post  of  cast-iron  will  with  safety  sustain. 

Mr.  Hodgkinson's  formula  for  the  breaking  strength  in  tons  of 
solid  pillars  of  cast-iron  in  the  case  of  pillars  with  rounded  ends 

is — 

/Z3* 
Strength  in  tons=14'9— ; 

and  in  pillars  with  flat  ends — 

Strength  in  tons=44'16^ 

where  d  is  the  diameter  in  inches,  and  I  the  length  in  feet. 

The  loads  in  cwts.  which  may  be  put  upon  solid  cylinders  or 
columns  of  cast-iron  of  different  diameters  and  lengths  are  ex- 
hibited in  the  following  table : — 


132 


MECHANICS    OF   THE    STEAM-ENGINE. 


WEIGHT  IN  CWTS.  SUSTAINABLE  WITH  SAFETY  BY  SOLID  CYL- 
INDERS OR  COLUMNS  OF  CAST-IRON  OF  DIFFERENT  DIAM- 
ETERS AND  LENGTHS. 


Diameter 

LENGTH  OP  COLTTMN  IK  FEET. 

of  column 

in  inches. 

6 

8 

10 

12 

14 

16 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

cwts. 

2 

61 

50 

40 

32 

26 

22 

2i 

105 

91 

77 

65 

55 

47 

3 

163 

145 

128 

111 

97 

84 

si 

232 

214 

191 

172 

156 

135 

4 

310 

288 

266 

242 

220 

198 

*i 

400 

379 

354 

327 

301 

275 

5 

601 

479 

452 

427 

394 

365 

6 

592 

573 

550 

525 

497 

469 

7 

1013 

989 

959 

924 

887 

848 

8 

1315 

1289 

1259 

1224 

1185 

1142 

In  hollow  pillars  nearly  the  same  laws  obtain  as  in  solid. 
Thus  in  the  case  of  hollow  pillars,  with  rounded  ends  or  movable 
ends,  like  the  cast-iron  connecting-rod  of  a  steam-engine,  the 

formula  is-                                          n^-tf* 
Strength  in  tons=13 _ 

and  in  the  case  of  hollow  pillars,  with  flat  ends — 

T)S'6 x73-6 

Strength  in  tons=44'3 . ; 


where  D  is  the  external  and  d  the  internal  diameter.  The 
strength  of  a  pillar  with  a  cross  section  of  the  form  of  a  cross 
was  found  to  be  only  about  half  as  great  as  that  of  a  cylindrical 
hollow  pillar.  It  was  also  found  that  in  pillars  of  the  same 
dimensions,  but  of  different  materials,  taking  the  strength  of 
cast-iron  at  1,000,  that  of  wrought-iron  was  1,745,  cast  steel 
2,518,  Dantzic  oak  108-8,  and  red  deal  78-5. 

Mr.  Hodgkinson's  rule  for  the  breaking  weight  of  cast-iron 
beams  is  as  follows : — 


STRENGTH    OF   CAST-IKON    BEAMS    AND    SHAFTS.        133 

STEENGTH   OF   CAST-IEON  BEAMS. 

KULE. — Multiply  the  sectional  area  of  the  bottom  flange  in 
square  inches  by  the  depth  of  the  learn  in  inches,  and  divide 
the  product  by  the  length  between  the  supports  also  in  inches. 
Then  514  times  the  quotient  will  be  the  breaking  weight  in 
cwt«. 

8TBEXGTH  OF  SHAFTS. 

44  Ibs.  acting  at  a  foot  radius  will  twist  off  the  neck  of  a 
shaft  of  lead  1  inch  diameter,  and  the  relative  strengths  of  other 
materials,  lead  being  1,  is  as  follows: — Tin,  1-4;  copper  4'3 ; 
yellow  brass,  4'6;  gun  metal,  5  ;  cast-iron,  9 ;  Swedish  iron,  9 '5 ; 
English  iron,  lO'l ;  blistered  steel,  16'6;  shear  steel,  17;  and 
cast  steel,  19'5.  The  strength  of  a  shaft  increases  as  the  cube 
of  its  diameter. 


CHAPTER  HI. 

THEORY  OP  THE  STEAM-ENGINE. 

THE  Steam-Engine  is  a  machine  for  extracting  mechanical 
power  from  heat  through  the  agency  of  water. 

Heat  is  one  form  of  mechanical  power,  or  more  properly,  a 
given  quantity  of  heat  is  the  equivalent  of  a  determinate  amount 
of  mechanical  power ;  and  as  heat  is  capable  of  producing  power, 
so  contrariwise  power  is  capable  of  producing  heat.  The  nature 
of  the  medium  upon  which  the  heat  acts  in  the  production  of 
the  power — whether  it  be  water,  air,  metal,  or  any  other  sub- 
stance— is  immaterial,  except  in  so  far  as  one  substance  may  be 
more  convenient  and  manageable  in  practice  than  another.  But 
with  any  given  extremes  of  temperature,  and  any  given  expen- 
diture of  heat,  the  amount  of  power  generated  by  any  given 
quantity  of  heat  will  be  the  same,  whatever  be  the  nature  of  the 
substance  on  which  the  heat  is  made  to  act  in  the  generation  of 
the  power.  And  just  in  the  proportion  in  which  power  is  gen- 
erated so  will  the  heat  disappear.  "We  cannot  have  both  the 
heat  and  the  power;  but  as  the  one  is  transformed  into  the 
other,  so  it  will  follow  that  the  acquisition  of  the  one  entails  a 
proportionate  loss  of  the  other,  and  this  loss  cannot  possibly  be 
prevented.  It  has  been  already  explained  that,  as  in  all  cases  in 
which  power  is  produced  in  a  steam-engine,  there  must  be  a  dif- 
ference of  pressure  on  the  two  sides  of  the  piston,  or  between  the 
boiler  and  the  condenser ;  so  in  all  cases  in  which  power  is  pro- 


NATURE   AND    EFFECTS   OF  HEAT.  135 

duced  in  any  species  of  caloric  engine,  there  must  be  a  difference 
of  temperature  between  the  source  of  heat  and  the  atmosphere 
or  refrigerator.  The  amount  of  this  difference  will  determine 
the  amount  of  power,  up  to  a  certain  limit,  which  a  unit  of 
heat  will  generate  in  any  given  engine.  But  as  it  has  been  al- 
ready explained  that  the  mechanical  equivalent  of  the  heat  con- 
sumed in  heating  1  Ib.  of  water  1°  Fahrenheit  would,  if  utilised 
without  loss,  raise  a  weight  of  772  Ibs.  1  foot  high,  it  will  fol- 
low that  in  no  engine  whatever  can  a  greater  performance  be 
obtained  than  this,  whatever  difference  of  temperature  we  may 
assume  between  the  extremes  of  heat  and  cold.  A  weight  of 
772  Ibs.  raised  1  foot  for  1°  Fahrenheit  is  equivalent  to  a  weight 
of  1389*6  Ibs.  raised  1  foot  for  1°  Centigrade;  and  for  conven- 
ience the  term  foot-pound  is  now  very  generally  employed  to  de- 
note the  dynamical  unit,  or  measure  of  power,  expressed  by  a 
weight  of  1  Ib.  raised  through  1  foot.  A  horse-power,  or  as  it 
is  now  commonly  termed  an  actual  or  indicator  horse-power — 
to  distinguish  it  from  a  nominal  horse-power,  which  is  a  mere 
measure  of  capacity — is  a  dynamical  unit  expressed  by  33,000  Ibs. 
raised  1  foot  high  in  a  minute ;  or  it  is  650  foot-pounds  per  sec- 
ond; 33,000  foot-pounds  per  minute;  or  1,980,000  foot-pounds 
per  hour.  This  unit  takes  into  account  the  rate  ofworTc  of  the 
machine. 

Heat,  like  light,  is  believed  to  be  a  species  of  motion,  and 
there  are  three  forms  of  heat  of  which  a  work  of  this  nature  re- 
quires to  take  cognisance — Sensible  Heat,  Latent  Heat,  and 
Speciffc  Heat. 

Sensible  Heat  is  heat  that  is  sensible  to  the  touch,  or  measur- 
able by  the  thermometer.  Latent  neat  is  the  heat  which  a  body 
absorbs  in  changing  its  state  from  solid  to  liquid,  and  from  liquid 
to  aeriform,  without  any  rise  of  temperature,  or  it  is  the  heat  ab- 
sorbed in  expansion.  And  Specific  Heat  is  an  expression  for  the 
relative  quantity  of  heat  in  a  body  as  compared  with  that  in 
some  other  standard  body  of  the  same  temperature.  There  is  a 
constant  tendency  in  hot  bodies  to  cool,  or  to  transfer  part  of 
their  heat  to  surrounding  colder  bodies ;  and  contiguous  bodies 
are  said  to  be  of  equal  temperatures  when  there  ceases  to  be  any 


136  THEORY   OF  THE   STEAM-ENGINE. 

transfer  of  heat  from  one  to  the  other.  The  most  prominent 
phenomena  of  heat  are  Dilatation,  Liquefaction,  and  Vaporisa- 
tion. 

Difference  between  temperature  and  quantity  of  heat. — It  is 
quite  clear  that  two  pounds  of  boiling  water  have  just  twice  the 
quantity  of  heat  in  them  that  is  contained  in  one  pound  of  boil- 
ing water.  But  it  does  not  by  any  means  follow,  nor  is  it  the 
case,  that  two  pounds  of  boiling  water  at  212°  contain  twice  the 
quantity  of  heat  that  is  contained  in  two  pounds  of  water  at 
106°.  Experiment  indeed  shows,  that  when  equal  quantities  of 
water  at  different  temperatures  are  mixed  together,  the  resulting 
temperature  is  the  mean  of  the  two,  so  that  if  a  pound  of  water 
at  200°  be  mixed  with  a  pound  of  water  at  100°,  we  have  a  re- 
sulting two  pounds  of  water  of  150°.  But  before  we  could  sup- 
pose that  a  pound  of  water  at  200°  has  twice  the  quantity  of  heat 
in  it  that  is  contained  in  a  pound  of  water  at  100°,  it  would  be 
necessary  to  conclude  that  water  at  0°  or  zero,  has  no  heat  in  at 
whatever.  This,  however,  is  by  no  means  the  case ;  and  tem- 
peratures much  below  zero  have  been  experimentally  arrived  at, 
and  even  naturally  occur  in  northern  latitudes.  A  pound  of  ice 
at  a  temperature  below  zero,  rises  in  temperature  by  each  suc- 
cessive addition  of  heat,  until  it  attains  the  temperature  of  32°, 
when  it  begins  to  melt ;  and,  notwithstanding  successive  addi- 
tions being  made  to  its  heat,  its  temperature  refuses  to  rise  above 
32°  until  liquefaction  has  been  completed.  So  soon  as  all  the  ice 
has  been  melted,  the  temperature  of  the  resulting  water  will 
continue  to  rise  with  each  successive  increment  of  heat,  until  the 
temperature  of  212°  has  been  attained,  when  the  water  will  boil, 
and  all  subsequent  additions  to  the  heat  will  be  expended  in  evap- 
orating the  water  or  in  converting  it  into  steam.  Although, 
therefore,  a  pound  of  water  in  the  form  of  steam  has  only  the 
same  temperature  as  a  pound  of  boiling  water,  it  has  a  great  deal 
more  heat  in  it,  as  is  shown  by  the  fact  that  it  will  heat  to  a 
given  temperature  a  great  many  more  pounds  of  cold  water  than 
a  pound  of  boiling  water  would  do. 

Absolute  zero. — The  foregoing  considerations  lead  naturally 
to  the  inquiry  whether,  although  bodies  at  the  zero  of  Fahren- 


DIFFEKENT   THERMOMETERS    COMPARED.  137 

heit's  scale  are  still  possessed  of  some  heat,  there  may  not,  never- 
theless, be  a  point  at  which  there  would  be  no  heat  whatever, 
and  which  point  therefore  constitutes  the  true  and  absolute  zero. 
Such  a  point  has  never  been  practically  arrived  at.  But  the  law 
of  the  elasticity  of  gases  and  their  expansion  by  heat,  leads  to 
the  conclusion  that  there  is  such  a  point,  and  that  it  is  situated 
461 '2°  Fahrenheit  below  the  zero  of  Fahrenheit's  scale,  or  in 
other  words  that  it  is  — 461 '2°  Fahrenheit,  — 274°  Centigrade, 
or  — 219'2°  Reaumur.  Mr.  Eankine  has  shown,  that  by  reckon- 
ing temperatures  from  this  theoretical  zero,  at  which  there  is  sup- 
posed to  be  no  heat  and  no  elasticity,  the  phenomena  dependent 
upon  temperature  are  more  readily  grouped  and  more  simply  ex- 
pressed than  would  otherwise  be  possible. 

Fixed  Temperatures. — The  circumstance  of  the  temperatures 
of  liquefaction  and  ebullition  being  fixed  and  constant,  enables 
us  to  obtain  certain  standard  or  uniform  temperatures,  to  which 
all  others  may  easily  he  referred.  One  of  these  standard  tem- 
peratures is  the  melting-point  of  ice,  and  another  is  the  boiling- 
point  of  pure  water  under  the  average  amospheric  pressure  of 
14*7  Ibs.  on  the  square  inch,  2116'8  Ibs.  on  the  square  foot ;  or  un- 
der the  pressure  of  a  vertical  column  of  mercury  29*922  inches 
high,  the  mercury  being  at  the  density  proper  to  the  tempera- 
ture of  melting  ice. 

Thermometers. — Thermometers  measure  temperatures  by  the 
dilatation  which  a  certain  selected  body  undergoes  from  the  appli- 
cation of  heat.  Sometimes  the  selected  body  is  a  solid,  such  as 
a  rod  of  brass  or  platinum ;  at  other  times  it  is  a  liquid,  such  as 
mercury  or  spirits  of  wine ;  and  at  other  tunes,  again,  it  is  a 
gas,  such  as  air  or  hydrogen.  In  a  perfect  gas  the  elasticity  is 
proportionate  to  the  compression,  whereas  in  an  imperfect  gas, 
such  as  carbonic  acid,  which  may  be  condensed  into  a  liquid,  the 
rate  of  elasticity  diminishes  as  the  point  of  condensation  is  ap- 
proached. Every  gas  approaches  more  nearly  to  the  condition 
of  a  perfect  gas  the  more  it  is  heated  and  rarefied,  but  an  abso- 
lutely perfect  gas  does  not  exist  in  nature.  Common  air,  how- 
ever, approaches  sufficiently  to  the  condition  of  a  perfect  gas,  to 
be  a  just  measure  of  temperatures  by  its  expansion. 


138  THEORY    OF   THE    STEAM-ENGINE. 

Air  and  all  other  gases  expand  equally  with  equal  increments 
of  temperature ;  and  it  is  found  experimentally  that  a  cubic  foot 
of  air  at  the  temperature  of  melting  ice,  or  32°,  will  form  1*365 
cubic  feet  of  the  same  pressure  at  the  temperature  of  boiling 
water,  or  212°.  Thermometers,  however,  are  not  generally  con- 
structed with  air  as  the  expanding  fluid,  except  for  the  measure- 
ment of  very  high  temperatures.  The  most  usual  species  of  ther- 
mometer consists  of  a  small  glass  bulb  filled  with  mercury,  and 
in  connection  with  a  capillary  tube.  The  bulb  is  immersed  in  the 
substance  the  temperature  of  which  it  is  desired  to  ascertain ; 
and  the  amount  of  the  dilatation  is  measured  by  the  height  to 
which  the  mercury  is  forced  up  the  capillary  tube.  The  ther- 
mometer commonly  used  in  this  country  is  Fahrenheit's  ther- 
mometer, of  which  the  zero  or  0  of  the  scale  is  fixed  at  the 
temperature  produced  by  mixing  salt  with  snow ;  and  which 
temperature  is  32°  below  the  freezing-point  of  water.  The  Cen- 
tigrade thermometer  is  that  commonly  used  on  the  continent  of 
Europe ;  and  it  is  graduated  by  dividing  the  distance  between  the 
point  where  the  mercury  stands  at  the  freezing-point  of  water, 
and  the  point  where  it  stands  at  the  boiling-point  of  water,  into 
100  equal  parts.  Of  this  thermometer  the  zero  is  at  the  freez- 
ing p.oint  of  water.  Another  thermometer,  called  Beaumur's 
thermometer,  has  its  zero  also  at  the  freezing-point  of  water ; 
and  the  distance  between  that  and  the  boiling-point  of  water 
is  divided  into  eighty  equal  parts.  Hence  80°  Eeaumur  are  equal 
to  100°  Centigrade,  and  180°  Fahrenheit.  The  correspond- 
ing degrees  of  these  thermometers  are  shown  in  the  following 
table : — 


DILATATION   OF   SOLIDS. 


139 


CEXTIGBADE,    EEATTMUR's,    AXD   FAHEEXHEIT's   THEEMOMETERS. 


Cent. 

Reau. 

Fahr. 

Cent 

Reau. 

Fahr. 

Cent. 

Reau. 

Fahr. 

Cent. 

Reau. 

Fahr. 

100 

so- 

212- 

64 

51-2 

147-2 

29 

23-2 

84-2 

—6 

—4-8 

21-2 

99 

79-2 

210-2 

63 

50-4 

145-4 

28 

22-4 

82-4 

7 

5-6 

19-4 

9S 

78-4 

20S-4 

62 

49-6 

143-6 

27 

21-6 

80-6 

8 

6-4 

17-6 

97 

77-6 

206-6 

61 

48-8 

141-8 

26 

20-8 

7S-8 

9 

7-2 

15-8 

96 

76-8 

204-8 

60 

48- 

140-   i 

25 

20- 

77- 

10 

8- 

14- 

95 

76- 

203- 

59 

47-2 

138-2 

24 

19-2 

75-2 

11 

8-8 

12-2 

94 

75-2 

201-2 

58 

46-4 

136-4 

23 

18-4 

73-4 

12 

9-6 

10-4 

93 

74-4 

199-4 

57 

45-6 

134-6 

22 

17-6 

71-6 

18 

10-4 

8-6 

92 

73-6 

197-6 

56 

44-8 

132-8 

21 

16-8 

69-8 

14 

11-2 

6-8 

91 

72-8 

195-8 

55 

44- 

131- 

20 

16- 

68- 

15 

12- 

5- 

90 

72- 

194- 

54 

43-2 

129-2 

19 

15-2 

66-2 

16 

12-8 

3-2 

89 

71-2 

192-2 

53 

42-4 

127-4 

18 

14-4 

64-4 

17 

13-6 

1-4 

88 

70-4 

190-4 

52 

41-6 

125-6 

17 

13-6 

62-6 

18 

14-4 

—0-4 

87 

69-6 

188-6 

51 

40-8 

123-8 

16 

12-8 

60-8 

19 

15-2 

2-2 

86 

68-8 

186-8 

50 

40- 

122- 

15 

12- 

59- 

20 

16- 

4- 

85 

68- 

185- 

49 

39-2 

120-2 

14 

11-2 

57-2 

21 

16-8 

5-8 

84 

67-2 

183-2 

43 

38-4 

118-4 

13 

10-4 

55-4 

22 

17-6 

7-6 

S3 

66-4 

181-4 

47 

37-6 

116-6 

12 

9-6 

53-6 

23 

18-4 

9-4 

82 

65-6 

179-6 

46 

36-8 

114-8 

11 

8-8 

51-8 

24 

19-2 

11-2 

81 

64-8 

177-8 

45 

36- 

118- 

10 

8- 

50- 

25 

20- 

13- 

80 

64- 

176- 

44 

35-2 

111-2 

9 

7-2 

48-2 

26 

20-8 

14-8 

79 

63-2 

174-2 

43 

34-4 

109-4 

8 

6-4 

46-4 

27 

21-6 

16-6 

78 

62-4 

172-4 

42 

33-6 

107-6 

7 

5-6 

44-6 

28 

22-4 

18-4 

77 

61-6 

170-6 

41 

82-8 

105-8 

6 

4-8 

42-8 

29 

23-2 

20-2 

76 

60-8 

168-8 

40 

82- 

104- 

5 

4- 

41- 

80 

24- 

22- 

75 

60- 

167' 

89 

81-4 

102-2 

4 

8-2 

89-2 

81 

24-8 

23-8 

74 

59-2 

165-2 

88 

30-2 

100-4 

8 

2-4 

87-4 

82 

25-6 

25-6 

73 

53-4 

163-4 

87 

29-6 

98-6 

2 

1-6 

85-6 

83 

26-4 

27-4 

72 

57-6 

161-6 

36 

28-8 

96-8 

1 

0-8 

83-8 

84 

27-2 

29-2 

71 

56-8 

159-8 

85 

23- 

95- 

0 

o- 

82- 

85 

28- 

81- 

70 

56- 

158- 

84 

27-2 

93-2 

_1 

—0-8 

80-2 

86 

28-8 

32-8 

69 

55-2 

156-2 

83 

26-4 

91-4 

2 

1-6 

28-4 

87 

29-6 

34-6 

68 

54-4 

154-4 

82 

25-6 

89-6 

3 

2-4 

26-6 

83 

80-4 

86-4 

67 

53-6 

152-6 

81 

24-8 

87-8 

4 

8-2 

24-8 

39 

81-2 

88-2 

66 

52-8 

150-8 

80 

24- 

86-  • 

5 

4- 

28- 

40 

82- 

40' 

65 

25- 

149- 

"Water,  in  common  with  molten  cast-iron,  molten  bismuth, 
and  various  other  fluid  substances,  the  particles  of  which  assume 
a  crystalline  arrangement  during  congelation,  suffers  an  increase 
of  bulk  as  the  point  of  congelation  is  approached,  and  expands  in 
solidifying.  But  so  soon  as  any  of  these  substances  has  become 
solid,  it  then  contracts  with  every  diminution  of  temperature. 
Water  in  freezing  bursts  by  its  expansion  any  vessel  in  which  it 
may  be  confined,  and  ice,  being  lighter  than  water,  floats  upon 
water.  So  also  for  a  like  reason  solid  cast-iron  floats  on  molten 
cast-iron.  The  point  of  maximum  density  of  water  is  39'1°  Fah- 
renheit, and  between  that  point  and  32°  the  bulk  of  water  in- 


140  THEORY   OF   THE    STEAM-ENGINE. 

creases  by  cold.  A  cubic  foot  of  water  at  32°  weighs  62-425 
Ibs.,  whereas  a  cubic  foot  of  ice  at  32°  weighs  only  57'5  Ibs. 
There  is  consequently  a  difference  of  nearly  5  Ibs.  in  each  cubic 
foot,  between  the  weight  of  ice  and  the  weight  of  water. 


DILATATION. 

Dilatation  of  Solids. — A  solid  body  of  homogeneous  texture 
will  dilate  uniformly  throughout  its  entire  bulk  by  the  applica- 
tion of  heat.  Thus,  if  it  be  found  that  a  bar  of  zinc  is  increased 
one  340th  part  of  its  length  by  being  raised  in  temperature  from 
32°  to  212°,  its  breadth  will  also  be  increased  one  340th  part, 
and  its  thickness  will  be  increased  one  340th  part.  It  is  found, 
moreover,  that  equal  increments  of  heat  produce  equal  augmen- 
tations of  volume  in  nearly  all  bodies,  at  all  temperatures,  until 
the  melting-point  is  approached,  when  irregularities  occur. 
Different  solids  dilate  to  different  amounts  when  subjected  to 
the  same  increase  of  temperature,  and  advantage  is  taken  of  this 
property  in  the  arts  in  the  construction  of  time-keepers  and 
other  instruments.  Thus,  in  Harrison's  gridiron  pendulum,  the 
ball  is  composed  of  bars  of  different  metals,  some  of  which  ex- 
pand more  than  the  others  at  the  same  temperature ;  and  as  the 
bars  which  expand  the  most  are  fixed  at  the  lower  ends  and  ex- 
pand upwards,  they  compensate  for  the  expansion  of  the  pendu- 
lum rod  in  the  opposite  direction,  and  maintain  the  centre  of  os- 
cillation in  the  same  place.  The  following  table  exhibits  the 
rates  of  dilatation  of  various  solids,  as  ascertained  by  the  best 
authorities : — 


DILATATION   PRODUCED    BY   HEAT. 


141 


DILATATION    OF   SOLIDS   BY   HEAT. 


Bodies. 

Dilatation  from  32°  to  212C,  according  t 
Flint  Glass  (English)     

Dilatation  in  Fractions. 

Decimal 
9  Lavoisier  and^ 

0-00081166 
0-00085655 
0-00087199 
0-00087572 
0-00089694 
0-00089760 
0-00091750 
0-00089089 
0-00107880 
0-00107915 
0-00107960 
0-00123956 
0-00122045 
0-00123504 
0-00146606 
0-00151361 
0-00155155 
0-00171220 
0-00171733 
0-00172240 
0-00186670 
0-00187821 
0-00188970 
0-00190868 
0-00190974 
0-00193765 
0-00217298 
0-00284836 

071. 

0-00083333 
0-00108333 
0-00115000 
0-00122500 
0-00125833 
0-00139167 
0-00170000 
0-00181667 
0-00187500 
0-00190833 

Vulgar. 
r^aplace. 

T^V? 

11  tfT 

± 

1  A  6 

TuTT 
ToStr 

•rrW 

± 

sis 
TOT 

Tl7 

li-ff 
-5$? 

F&T 
F4T 
Stt 

ttZ 
T5T 

6^S 
635" 

-sfe 
vb 

?tr 
Ti-s 

T5"0" 

lir 

T«W 
TTTT 

^ 
T5ff 

it* 
T$-5 

"5^5 
*4lT 
TTlbf 

Platinum  (accordin**  to  Borda)  

Glass  (French)  with  lead  

Glass  tube  without  lead  ..       

Ditto      

Ditto  

Ditto    

Glass  (St  Gobain)-  •  •  •  

Steel  (untempered)     

Ditto  

Ditto     

Steel  (yellow  temper)  annealed  at  65°  

Iron,  round  wire-drawn  

Gold  

Gold  (French  standard)  annealed  

Gold  (ditto)  not  annealed  

Ditto  

Ditto  

Ditto  

Silver  (French  standard)  

Silver  

Tin  Falmouth  

According  to  Smeai 

Glass,  white  (barometer  tubes)  \ 
Steel  

Steel  (tempered).  

Iron  

Copper.  

Copper  8  parts,  tin  1  

Brass  16  parts,  tin  1  

142 


THEOKY    OF   THE    STEAM-ENGINE. 


DILATATION   OF   SOLIDS   BY   HEAT — Continued. 


Bodies. 


Dilatation  in  Fractions. 


Decimal. 

Brass  wire 0-00193333 

Telescope  speculum  metal 0-00193333 

Solder  (copper  2  parts,  zinc  1) 0-00205833 

Tin  (fine) 0-00228333 

Tin(grain) 0'00248333 

Solder  white  (tin  1  part,  lead  2) 0-00250533 

Zinc  8  parts,  tin  1,  slightly  forged 0'00269167 

Lead 0-00286667 

Zinc 0-00294167 

Zinc  lenthened  -^  by  hammering 0-00310833 

Palladium  ( Wollaston) O'OOl  00000 

According  to  Dulong  and  Petit. 

P1  .._ „  j  32°  to  212°1  0-00088420 

Platmum \  32°  to  572°  0-00275482 

32°  to  212°  0-00086133 

Glass 32°  to  392°  0-00184502 

32°  to  572°  0-00303252 

T  32°  to  212°  0-00118210 

on 32°  to  572°  0-00440528 

r  (  32°  to  212°  0-00171820 

' '  (  32°  to  572°  0-00564&72 

According  to  Troughton. 

Platinum 0-00099180 

Steel 0-001 18990 

Steel  wire,  drawn 0-00144010 

Copper 0-00191880 

Silver j  0'00208260 

From  32°  to  217°  according  to  Roy. 

Glass  (tube) 0-00077550 

Glass  (solid  rod) 0-00080833 

Glass  cast  (prism  of) O'OOlllOOO 

Steel(rod  of) 0-00114400 

Brass  (Hamburg) 0-00185550 

Brass  (English)  rod 0-00189296 

Brass  (English)  angular 0-00189450 


DILATATION   PRODUCED    BY   HEAT.  143 

Measure  of  the  Force  of  Dilatation. — The  force  with  which 
solid  bodies  dilate  and  contract  is  equal  to  that  which  would 
compress  them  through  the  space  they  have  dilated,  or  to  that 
which  would  stretch  them  through  a  space  equal  to  the  amount 
of  their  contraction.  Now,  as  it  has  been  shown  to  be  a  phys- 
ical law  that  in  every  substance  whatever,  the  same  expenditure 
of  heat,  with  the  same  extremes  of  temperature,  will  generate 
the  same  amount  of  mechanical  power,  it  will  follow  that  the 
less  a  body  expands  with  any  given  increase  of  temperature,  the 
more  forcible  will  be  the  expansion,  since  the  force,  multiplied 
by  the  space  passed  through,  must,  in  every  case  be  a  constant 
quantity. 

Dilatation  of  Liquids. — The  rate  of  expansion  of  liquids 
becomes  greater  as  the  temperature  becomes  higher,  so  that  a 
mercurial  thermometer,  to  be  accurately  graduated,  should  have 
the  graduations  at  the  top  of  the  scale  somewhat  larger  than  at 
the  bottom.  It  so  happens,  however,  that  there  is  a  similar 
irregularity  in  the  expansion  of  the  glass  bulb,  but  in  an  opposite 
direction ;  and  one  error  very  nearly  corrects  the  other.  Ther- 
mometers are  accordingly  graduated  by  immersing  the  bulb  in 
melting  ice,  and  marking  the  point  at  which  the  mercury  stands. 
The  point  at  which  the  mercury  stands  when  the  bulb  is  im- 
mersed in  boiling  water  is  next  marked,  and  the  space  between 
the  two  marks  is  divided  into  180  equal  parts,  and  the  graduation 
is  extended  above  the  boiling-point  and  below  the  freezing,  by 
continuing  the  same  lengths  of  division  on  the  scale.  The 
increment  of  volume  which  water  receives  on  being  raised  from 
32°  to  212°  is  aVrd.  of  its  bulk  at  32°.  Mercury  at  32°  expands 
/,th  of  its  bulk  at  32°  by  being  raised  to  212° ;  and  alcohol,  by 
the  same  increase  of  temperature,  increases  in  volume  ^th  of  its 
bulk  at  32°. 

Compression  and  Dilatation  of  Oases. — When  a  gas  or 
vapour  is  compressed  into  half  its  original  bulk,  its  pressure  is 
doubled ;  when  compressed  into  a  third  of  its  original  bulk,  its 
pressure  is  trebled ;  when  compressed  into  a  fourth  of  its  original 
bulk,  its  pressure  is  quadrupled ;  and  generally  the  pressure  varies 
inversely  as  the  bulk  into  which  the  gas  is  compressed.  So,  in 


144 


THEORY   OF   THE   STEAM-ENGINE. 


like  manner,  if  the  volume  be  doubled,  the  pressure  is  made  one- 
half  of  what  it  was  before — the  pressure  being  in  every  case 
reckoned  from  0,  or  from  a  perfect  vacuum.  Thus,  if  we  take 
the  average  pressure  of  the  atmosphere  at  14'T/  Ibs.  on  the 
square  inch,  a  cubic  foot  of  air,  if  suffered  to  expand  into  twice 
its  bulk  by  being  placed  in  a  vacuum  measuring  two  cubic  feet, 
will  have  a  pressure  of  7*35  Ibs.  above  a  perfect  vacuum,  and 
also  of  7-35  Ibs.  below  the  atmospheric  pressure ;  whereas,  if  the 
cubic  foot  be  compressed  into  a  space  of  half  a  cubic  foot,  the 
pressure  will  become  29'4  Ibs.  above  a  perfect  vacuum,  and  14'7 
Ibs.  above  the  atmospheric  pressure.  This  law,  which  was  first 
investigated  by  Mariotte,  is  called  Nanette's  law.  It  has  already 
been  stated  that  a  cubic  foot  of  air  at  32°  becomes  1*365  cubic 
feet  at  212°,  the  pressure  remaining  constant;  or  if  the  volume 
be  kept  constant,  then  the  pressure  of  one  atmosphere  at  32°  be- 
comes 1-365  atmospheres,  or  a  little  over  1J  atmospheres  at  212°. 
These  two  laws,  which  are  of  the  utmost  importance  in  all  phys- 
ical researches,  it  is  necessary  fully  to  understand  and  remember. 
The  rates  of  dilatation  and  compression  for  each  gas  are  not  pre- 
cisely the  same ;  but  the  departure  from  the  law  is  so  small  as 
lo  be  practically  inappreciable.  According  to  M.  Eegnault,  the 
dilatation  under  the  same  pressure,  and  the  increase  of  pressure 
with  the  same  volume  of  different  gases  when  heated  from  32° 
to  212°,  is  as  follows : — 

00 -EFFICIENTS   OF  DILATATION  OF  DIFFERENT  GASES. 


Pressure 
under  constant 

voume. 

Dilatation 
under  constant 
pressure. 

Hydrogen.  . 

0-3667 

0-3661 

Atmospheric  air  

0-3665 

0-3670 

Nitrogen  

0-3668 

u 

0-3667 

0-3669 

Carbonic  acid  

0-3688 

0-3710 

Protoxide  of  nitrogen  

0-3676 

0.3719 

0-3845 

0-3903 

0-3829 

0.3877 

The  rates  of  dilatation  vary  somewhat  with  the  pressure  and 
temperature,  and  in  the  case  of  gases,  which  are  more  easily 


DILATATION   PRODUCED    BY   HEAT.  145 

condensable  into  liquids,  the  rate  of  dilatation  increases  rapidly 
with  the  density  ;  whereas  the  effect  of  heat  is  to  remove  these 
irregularities,  and  to  maintain  more  completely  the  condition  of 
a  perfect  gas. 

If  we  take  the  dilatation  of  atmospheric  air  when  heated  180°, 
or  from  32°  to  212°,  at  0-367  as  determined  by  M.  Eegnault, 
then  the  amount  of  expansion  which  it  will  undergo  from  each 
increase  of  one  degree  in  temperature  will  be  180th  of  0-367  = 
180th  of  ^V  —  188oVo<r  =  T&-5'  1&  otlier  words,  air  will  be 
enlarged  ^^th  part  of  its  bulk  at  32°  by  being  raised  one  degree 
in  temperature. 

If  the  same  quantity  of  air  or  gas  be  simultaneously  submitted 
to  changes  of  temperature  and  pressure,  the  relations  between 
its  volumes,  pressures,  and  temperatures,  will  be  expressed  by 
the  general  formula  — 


V        490  ± 

where  T  and  x'  express  the  number  of  degrees  above  or  below 
32°  at  which  the  temperature  stands,  +  being  used  when  above 
and  —  when  below  32°,  and  the  pressures  being  expressed  in  the 
usual  manner  by  P  and  p'.  By  this  formula,  the  volume  of  a  gas 
at  any  proposed  temperature  and  pressure  may  be  found,  if  its 
volume  at  any  other  temperature  and  pressure  be  given,  or  the 
same  thing  may  be  done  by  the  following  rule  :  — 

THE  BULK  OF  A  GAS  AT  32°  BEING  KNOWN,  TO  DETERMINE  ITS 
BULK  AT  ANT  OTHEB  TEMPEBATUBE,  THE  PHESSUBE  BEING 
CONSTANT. 

RULE.  —  Divide  the  difference  between  the  number  of  degrees  in 
the  temperature  and  32°  by  490.  Add  the  quotient  to  1  if 
the  temperature  be  above  32°,  and  subtract  it  from  I  if  it  be 
below  32°.  Multiply  the  volume  of  the  gas  at  32°  by  the 
resulting  number,  and  the  product  will  be  the  volume  of  the 
gas  at  the  proposed  temperature. 

Example  1.  —  What  volume  will  1000  cubic  inches  of  air  at 
82°  acquire  by  being  heated  to  1000°  Fahrenheit? 
7 


146 


THEORY   OF   THE   STEAM-ENGINE. 


EXPANSION  OF  DRY  AIR  BY  HEAT. 

[In  the  column!  V.  of  the  following  table  are  expressed  in  cubic  inches  the  volumes  which  a  thou- 
sand cubic  inches  of  air  at  328  will  have  at  the  temperatures  expressed  in  the  columns  T.,  the  air  being 
supposed  to  be  maintained  under  the  same  pressure.] 


T. 

V. 

T. 

V.      NT. 

V. 

T. 

V. 

T. 

T. 

—50 

832-7 

8 

951-0 

66 

1069.4 

124 

1187-8 

182 

1806-1 

-^9 

834-7 

9 

953-1 

67 

1071-4 

125 

1189-8 

183 

1308-2 

—48 

836-7 

10 

955-1 

68 

1073-5 

126 

1191-8 

184 

1310-2 

-47 

838-8 

11 

957-1 

69 

1075-5 

127 

1193-9 

185 

1812-2 

-46 

840-8 

12 

959-2 

70 

1077-6 

128 

1195-9 

186 

1314-3 

—45 

842-8 

13 

961-2 

71 

1079-6 

129 

1198-0 

187 

1316-3 

—44 

844-9 

14 

963-3 

72 

1081-6 

130 

1200-0 

188 

1318-4 

—43 

846-9 

15 

965-3 

73 

1083-7 

131 

1202-0 

189 

1320-4 

—42 

849-0 

16 

967-3 

74 

1085-7 

132 

1204-1 

190 

1822-4 

-^1 

851-0 

17 

969-4 

75 

1087-8 

133 

1206-1 

191 

1824-5 

-40 

853-1 

18 

971-4 

76 

1089-8 

134 

1208-2 

192 

1326-5 

—89 

855-1 

19 

973-5 

77 

1091-8 

135 

1210-2 

193 

1828-6 

—88 

857-1 

20 

975-5 

78 

1093-9 

136 

1212-2 

194 

1380-5 

—37 

859-2 

21 

977-6 

79 

1095-9 

187 

1214-8 

195 

1332-6 

—86 

861-2 

22 

979-6 

80 

1098-0 

188 

1216-8 

196 

1834-7 

—35 

868-8 

23 

981-6 

81 

1100-0 

139 

1218-4 

197 

1836-7 

—  S4 

865-3 

24 

988-7 

82 

1102-0 

140 

1220-4 

198 

1388-8 

-88 

867-3 

25 

985-7 

83 

1104-1 

141 

1222-4 

199 

1340-8 

—32 

869-4 

26 

987-8 

84 

1106-1 

142 

1224-5 

200 

1842-9 

—31 

871-4 

27 

989-8 

85 

1108-2 

143 

1226-5 

201 

1344-9 

—30 

878-5 

28 

991-8 

86 

1110-2 

144 

1228-6 

202 

1346-0 

—29 

875-5 

29 

993-9 

87 

1112-2 

145 

1230-6 

203 

1849-0 

—28 

877-6 

SO 

995-9 

88 

1114-3 

146 

1232-7 

204 

1351-1 

—27 

879-6 

81 

998-0 

89 

1116-8 

147 

1234-7 

205 

1853-1 

—26 

881-6 

82 

looo-o 

90 

11184 

148 

1236-7 

206 

1855-1 

—25 

883-7 

33 

1002-0 

91 

1120-4 

149 

1288-8 

207 

1357-3 

—24 

885-7 

34 

1004-1 

92 

1122-4 

150 

1240-8 

208 

1359-3 

—23 

887-8 

85 

1006-1 

98 

1124-5 

151 

1242-9 

209 

1861-3 

—22 

889-8 

36 

1008-2 

94 

1126-5 

152 

1244-9 

210 

1863-4 

—21 

891-8 

37 

1010-2 

95 

1128-6 

153 

1246-9 

211 

1865-5 

-20 

893-9 

88 

1012-2 

96 

1130-6 

154 

1249-0 

212 

1867-6 

—19 

895-9 

89 

1014-3 

97 

1132-7 

155 

1251-0 

213 

1369-2 

—18 

898-0 

40 

1016-3 

98 

1134-7 

156 

1258-0 

214 

1371-4 

—17 

900-0 

41 

1018-4 

99 

1136-7 

157 

12551 

215 

1373-2 

—16 

902-0 

42 

1020-4 

100 

1188-8 

158 

1257-1 

216 

1375-5 

—15 

904-1 

43 

1022-4 

101 

1140-8 

159 

1259-2 

217 

1377-5 

—14 

906-1 

44 

1024-5 

102 

1142-0 

160 

1261-2 

218 

1879-6 

—13 

908-2 

45 

1026-5 

103 

1144-9 

161 

1268-8 

219 

1381-6 

—12 

910-2 

46 

1028-6 

104 

1147-0 

162 

1265-3 

220 

1883-7 

—11 

912-2 

47 

1030-6 

105 

1149-0 

162 

1267-3 

230 

1404-1 

—10 

914-8 

48 

1032-7 

106 

1151-0 

164 

1269-4 

240 

1424-5 

9 

916-3 

49 

1034-7 

107 

1153-1 

165 

1271-4 

250 

1444-9 

—  8 

918-4 

50 

1086-7 

108 

1155-1 

166 

1273-5 

260 

1465-3 

—  7 

920-4 

51 

1088-8 

109 

1157-1 

167 

1275-5 

270 

1485-7 

—  6 

922-5 

52 

1040-8 

110 

1159-2 

168 

1277-5 

280 

1506-1 

—  5 

924-5 

53 

1042-9 

111 

1161-2 

169 

1279-6 

290 

1526-5 

—  4 

926-5 

54 

1044-9 

112 

1163-3 

170 

1281-6 

300 

1546-9 

—  3 

928-6 

55 

1046-9 

113 

1165-3 

171 

1283-7 

400 

1751-0 

—  2 

930-6 

56 

1049-0 

114 

1167-8 

172 

1285-7 

500 

1955-1 

j 

932-7 

57 

1051-0 

115 

1169-4 

173 

1287-8 

600 

2159-2 

0 

984-7 

68 

1058-1 

116 

1171-4 

174 

1289-8 

700 

2368-8 

1 

936-7 

59 

1055-1 

117 

1173-5 

175 

1291-8 

800 

2567-3 

2 

988-8 

60 

1057-1 

118 

1175-5 

176 

1293-9 

900 

2771-4 

8 

940-8 

61 

1059-2 

119 

1177-6 

177 

1295-9 

1000 

2975-5 

4 

942-9 

62 

1061-2 

120 

1179-6 

178 

1298-0 

1500 

8997-9 

5 

944-9 

63 

1063-3 

121 

1181-3 

179 

1300-0 

2000 

5016-3 

6 

947-0 

64 

1065-3 

122 

1188-7 

180 

1802-0 

2500 

6036-7 

7 

949-0 

65 

1067-3 

128 

1185-7 

181 

1304-1 

8000 

7057-1 

DILATATION   PRODUCED    BY   HEAT.  147 

The  difference  between  1000°  and  32°  is  968,  which  divided  by 
490  =  1-9755,  and  this  added  to  1  —  2*9755.  Then  1000  x  2-9755 
=  2975-5,  which  will  be  the  volume  in  cubic  inches  at  1000°. 

Example  2.— What  will  be  the  volume  of  the  above  air  at  2000°  ? 

Here  2000  —  32  =  1968  which  -j-  by  490  •=  4-0163,  and  this 
added  to  1  =  5-0163.  Finally,  5-0163  x  1000  =  5016-3,  which 
will  be  the  volume  of  the  air  in  cubic  inches  at  2000°. 

The  volume  which  1000  cubic  inches  of  air  at  32°  acquires  at 
all  the  various  temperatures  between  —  50°  and  3000°  is  shown 
in  the  preceding  table : 

ANOTHER  ETJLE. — To  each  of  the  temperatures  'before  and  after 
expansion  add  the  constant  number  459 :  divide  the  greater 
sum  by  the  lesser,  and  multiply  the  quotient  by  the  volume 
at  the  lower  temperature,  and  the  product  will  give  the  ex- 
panded volume. 
Example  1. — What  will  be  the  volume  of  1000  cubic  inches 

of  air  at  32°  when  heated  to  212°,  the  pressure  being  without 

alteration  ? 

212  +  459 
Here       32  +  459"  =  !'366>  which  multiplied  by  1000=1366, 

which  will  be  the  volume  in  cubic  inches  at  212°. 

Example  2. — If  the  volume  of  steam  at  212°  be  1696  tunes 
the  volume  of  the  water  which  produced  it,  what  will  the  vol- 
ume be  if  the  steam  be  heated  to  250-3  degrees  Fahrenheit,  the 
pressure  remaining  constant  ? 

Here  by  the  rule  212+459=671  and  250-3+459=709-3°. 
Moreover,  709-3  divided  by  671  and  multiplied  by  1696=1792-8, 
which  will  be  the  bulk  which  the  1696  measures  of  steam  will 
acquire  when  heated  to  250-3°  out  of  contact  with  water,  the 
pressure  remaining  the  same  as  at  first. 

If  we  take  the  co-efficient  of  expansion  of  a  perfect  gas  be- 
tween 32°  and  212°  at  0-365  instead  of  0'367,  the  expansion  per 
degree  Fahrenheit  will  be  ^.3-  of  the  total  bulk=0-0020276 
per  degree  Fahrenheit,  instead  of  ^J^th,  as  supposed  by  the  rule 
from  which  the  table  is  computed.  This  is  equivalent  to  start- 
ing from  the  point  of  absolute  zero,  or  461-2°  below  the  zero  of 
Fahrenheit;  as  461-20  +  320=493'25 


148 


THEORY    OF   THE    STEAM-ENGINE. 


TABLE  SHOWING  THE  MELTING  POINTS  OF  VARIOUS  BODIES,  IN  DE- 
GEEES  OF  FAHRENHEIT'S  THERMOMETER. 


Name  of  Substance. 


Degrees  Fahren.     Experimentalist. 


Platinum 3082° 

English  wrought-iron 2912 

French        "          "     2732 

Steel 2552 

"    another  sample 2372 

Cast-iron 2192 

manganese 2282 

brown,  fusible 2192 

very  fusible 2012 

white,  fusible -.          2012 

very  fusible 1922 

Gold  (very  pure) 2282 

Gold  coin 2156 

Copper 1922 

Brass 1859 

Silver  (very  pure) 1832 

Bronze 1652 

Antimony 810 

700 

Zinc 705 

680 
629 

Lead 608 

590 
518 

5?5 
480 
512 
455 

Tin 446 

442 
433 
Alloy,  5  parts  tin  „„, 

1  part  lead      ' ' 
Alloy,  4  parts  tin  oho 

'  1  part  lead      ' ' 
Alloy,  3  parts  tin  „„,- 

1  part  lead      ' ' 
Alloy,  2  parts  tin  .  Rrt 

1  part  lead      

Alloy,  1  part  tin     [  652 

3  parts  lead  f 


Clarke. 

Vauquelin. 

Pouillet. 


Daniell. 
PouiUet. 


Murray. 
G.  Morveau. 

Pouillet. 

Person. 

Pouillet. 

Irvine. 

Person. 

Ermann. 

Pouillet. 

Crichton. 
G.  Morveau. 

Person. 

PouiUet. 

Crichton. 

Ermann. 

Pouillet. 


MELTING  POINTS   OF   SOLIDS. 


149 


TABLE  SHOWING   THE   MELTING  POINTS  OF  VAEIOTTS  BODIES,  IN  DE- 

GEEES  OF  FAHRENHEIT'S  THERMOMETER — continued. 


Name  of  Substance. 

Degrees  Fahren. 

Experimentalist. 

Alloy,  3  parts  tin 

392 

Pouillet 

1  part  bismuth 
Alloy,  2  parts  tin 

333-9 

« 

1  part  bismuth        *     "    * 
Alloy,  1  part  tin 

286-2 

« 

1  part  bismuth 
Alloy,  4  parts  tin           ) 
1  part  lead           >•  

246 

« 

6  parts  bismuth  ) 

239 

Person. 

237 
225 

Dumas. 
Pouillet. 

Alloy,  2  parts  lead         ) 
3  parts  tin            >•  

212 

M 

5  parts  bismuth  ) 
Alloy,  5  parts  lead 
3  parts  tin            •  

212 

M 

8  parts  bismuth 
Alloy,  1  part  lead 
1  part  tin 

201 

(( 

4  parts  bismuth 
Soda  

194 

Ga.y-Lus8ac 

Potash  

162 

M 

136 
111-6 
109 

Pouillet 
Person. 
Pouillet. 

100 
158 

Murray. 
Pouillet. 

164 

M 

Wax  unbleached  

142 

«( 

Stearine  .  .  .  .  ,  

143 
120 

Person. 
Pouillet 

Spermaceti  

109 
120 

M 

Acetic  acid  

113 

<( 

Tallow  

92 

(c 

Ice  

32 

« 

Oil  of  turpentine  

14 

cc 

—  38.2 

{( 

150  THEOEY   OF  THE   STEAM-ENGINE. 

LIQUEFACTION. 

Solidity  is  an  accident  of  temperature,  as  there  is  every  rea- 
son to  believe  that  there  is  no  substance  in  nature  which  may 
not  be  melted,  and  even  vaporised,  by  the  application  of  power- 
ful heat. 

There  are  two  incidents  attending  liquefaction  that  are  wor- 
thy of  special  attention :  the  first  is  that  the  liquefaction  al- 
ways takes  place  at  the  same  temperature  in  the  case  of  the 
same  substance,  so  that  the  melting-point  may  in  fact  be  used 
as  an  index  of  temperature ;  and  the  second  is  that  during  lique- 
faction the  temperature  remains  fixed,  the  accession  of  heat 
which  has  been  received  during  the  process  of  liquefaction  being 
consumed  or  absorbed  in  accomplishing  the  liquefaction,  or  in 
other  words  it  has  become  latent.  This  heat  is  given  out  again 
in  the  process  of  solidification.  "Water  deprived  of  air  and  cov- 
ered with  a  thin  film  of  oil  may  be  cooled  to  20°  or  22°  below 
the  freezing-point.  But  on  solidification  the  temperature  will 
rise  to  the  freezing-point.  Each  different  substance  has,  under 
ordinary  circumstances,  its  own  particular  melting-point;  but 
it  is  found  that  the  electrical  condition  of  a  body  affects  its  melt- 
ing-point, and  that  electricity  will  fuse  bodies  at  a  low  tempera- 
ture which  commonly  require  for  their  fusion  a  very  high  degree 
of  heat.  Tims,  platinum  may  be  melted  or  vaporised  by  an 
electrical  current,  even  although  the  heat  generated  is  small ; 
and  a  process  for  separating  metals  from  their  ores  by  the  aid  of 
electricity  has  been  projected  by  using  low  temperatures,  aided 
by  electricity,  instead  of  high  degrees  of  heat.  In  Part  XV.  of 
Taylor's  Scientific  Memoirs,  page  432,  there  is  a  paper  '  On  the 
Incandescence  and  Fusion  of  Metallic  Wires  by  Electricity,'  by 
Peter  Riess,  being  the  substance  of  a  paper  read  before  the 
Eoyal  Society  of  Berlin ;  and  in  this  paper  it  is  shown  that 
electrical  fusion  and  vaporisation  may  take  place  at  tempera- 
tures far  below  those  at  which  metals  are  red  hot.  This  prop- 
erty of  electricity  promises  to  be  of  service  in  the  arts  both  in 
rendering  refractory  bodies  fusible  and  in  enabling  bodies  to  be 
melted  at  low  temperatures,  which  might  be  injured  in  their 
qualities  by  a  subjection  to  high  degrees  of  heat.  Thus  wrought- 


LATENT  HEAT  OF  LIQUEFACTION. 


151 


iron  if  heated  to  a  very  high  temperature,  is  liable  to  be  burnt, 
unless  carefully  preserved  from  contact  with  the  air ;  whereas 
by  sending  a  current  of  electricity  through  it,  fusion  may  be 
accomplished  at  a  comparatively  low  temperature,  and  any  in- 
jury to  the  metal  may  thus  be  prevented.  The  melting-points 
of  some  of  the  most  important  substances  are  given  in  the  pre- 
ceding table. 

Latent  Heat  of  Liquefaction. — Ice  in  melting  absorbs  as 
much  heat  as  would  raise  the  temperature  of  the  same  weight 
of  water  142-65°,  or  as  would  raise  142-65  times  that  weight  of 
water  1  degree ;  yet,  notwithstanding  this  accession  of  heat,  the 
ice,  during  liquefaction,  does  not  rise  above  32°.  If  the  heat 
employed  to  melt  ice  was  applied  to  heat  the  same  weight  of 
ice-cold  water,  it  would  heat  it  to  the  temperature  of  142-65°  + 
32° =174-65°.  The  following  table  shows  the  amount  of  heat 
which  becomes  latent  in  the  liquefaction  of  various  bodies — the 
unit  of  latent  heat  being  the  amount  of  heat  necessary  to  raise 
the  same  weight  of  water  1  degree : 

TABLE  SHOWING  THE  HEAT  WHICH  BECOMES  LATENT  IN  THE 
LIQUEFACTION  OF  VAEIOUS  SOLID  BODIES,  AS  ASCERTAINED  BY 
M.  PEBSON. 


Names  of  Substances. 

Points  of 
Fusion 
Fahrenheit 

Latent  Heat 
for  Unity  of 
Weight 

Chloride  of  lime  

83-3 

72-42 

Phosphate  of  soda  

97-5 

120-24 

Phosphorus  

111-6 

8-48 

Bees'-wax  (yellow)  

143-6 

78-32 

D'Arcet's  alloy  

204-8 

10-73 

239-0 

16-61 

Tin  

456-0 

26-74 

Bismuth  

518-0 

22-32 

Nitrate  of  soda  

590-9 

113-36 

Lead  

629-6 

9-27 

Nitrate  of  potash  

642.2 

83-12 

7934 

49-43 

By  this  table  we  see  that  the  heat  which  becomes  latent  in 
melting  a  pound  of  bees'  wax  would  raise  the  temperature  of  a 


152  THEORY   OF  THE   STEAM-ENGINE. 

pound  of  water  78'32  degrees;  and  the  heat  which  becomes 
latent  in  melting  a  pound  of  lead  would  raise  the  temperature 
of  a  pound  of  water  9-27  degrees. 

When  there  is  no  external  source  of  heat,  from  which  the 
heat  which  becomes  latent  in  liquefaction  can  be  derived,  and 
the  circumstances  are,  nevertheless,  such  as  to  cause  liquefaction 
to  take  place,  the  heat  which  becomes  latent  is  derived  from  the 
substances  themselves,  and  correspondingly  lowers  their  temper- 
atures. Thus,  when  snow  and  salt  are  mixed  together,  the 
snow  and  salt  are  dissolved.  But,  as  in  melting  they  absorb 
heat,  and  as  there  is  no  external  source  from  which  the  heat  is 
derived,  the  temperature  of  the  mixture  falls  very  much  below 
that  of  either  of  the  substances  before  mixing.  So,  also,  when 
saltpetre  and  other  salts  are  dissolved  in  water,  cold  is  produced, 
and  on  this  principle  the  freezing  mixtures  are  compounded 
which  are  employed  to  produce  artificial  cold  in  warm  climates. 
A  more  effectual  process,  however,  is  to  compress  air,  which 
heats  it ;  and  the  superfluous  heat  being  got  rid  of  by  water,  it 
will  follow  that  when  this  air  is  again  expanded,  it  will  take 
back  an  amount  of  heat  equal  to  that  which  it  before  lost,  and 
which  demand  for  heat  may  be  made  to  cool  surrounding  bodies. 
A  very  effectual  freezing  machine  is  constructed  on  this  princi- 
ple. But  it  is  material  that  the  air  in  expanding  should  be  made 
to  generate  power,  else  the  friction  consequent  on  its  escape  will 
generate  heat. 

VAPORISATION. 

As  the  first  phenomenon  of  the  application  of  heat  to  a  solid 
substance  is  to  dilate  it,  and  the  next  to  melt  it,  so  also  the  fur- 
ther application  of  heat  converts  it  from  a  liquid  into  a  vapour 
or  gas.  The  point  at  which  successive  increments  of  heat,  in- 
stead of  raising  the  temperature,  are  absorbed  in  the  generation 
of  vapour,  is  called  the  boiling-point  of  the  liquid.  Different 
liquids  have  different  boiling-points  under  the  same  pressure, 
and  the  same  liquid  will  boil  at  a  lower  temperature  in  a  va- 
cuum, or  under  a  low  pressure,  than  it  will  do  under  a  high 
pressure.  As  the  pressure  of  the  atmosphere  varies  at  different 


LATENT    HEAT    OF   VAPORISATION. 


153 


altitudes,  liquids  will  boil  at  different  temperatures  at  different 
altitudes,  and  the  height  of  a  mountain  may  be  approximately  de- 
termined by  the  temperature  at  which  water  boils  at  its  summit. 

Difference  between  Gases  and  Vapours. — Vapours  are  sat- 
urated gases,  or  gases  are  vapours  surcharged  by  heat.  Ordi- 
nary steam  is  the  saturated  vapour  of  water,  and  if  any  of  the 
heat  be  withdrawn  from  it,  a  portion  of  the  water  is  necessarily 
precipitated.  This  is  not  so  in  the  case  of  a  gas  under  ordinary 
conditions.  But  if  the  gas  be  forced  into  a  very  small  bulk,  so 
that  much  of  the  heat  is  squeezed  out  of  it,  then  it  will  follow 
that  any  diminution  of  the  temperature  will  cause  a  portion  of 
the  gas  to  condense  into  a  liquid.  Surcharged  or  superheated 
steam  resembles  gas  in  its  qualities,  and  a  portion  of  the  heat 
may  be  withdrawn  from  such  steam,  without  producing  the  pre- 
cipitation of  any  part  of  its  constituent  water. 

Liquefaction  of  the  gases. — Many  of  the  gases  have  already 
been  brought  into  the  liquid  state,  by  the  conjoint  agency  of  cold 
and  compression,  and  all  of  them  are  probably  susceptible  of  a 
similar  reduction  by  the  use  of  means  sufficiently  powerful  for 
the  required  end.  They  must,  consequently,  be  regarded  as  the 
superheated  steams,  or  vapours,  of  the  liquids  into  which  they 
are  compressed.  The  pressures  exerted  by  some  of  these  steams 
or  gases  are  given  in  the  following  table : — 

TABLE    SHOWING    THE    TEMPEEATUBE    AND    PRESSURE    AT   WHICH 
THE   8EVEEAL   GASES   NAMED   ABE   LIQUEFIED. 


Name*  of  Gues  condensed. 

Temperature 
in  degrees 
Fahrenheit 

Pressure  ID 

Atmospheres. 

Temperature 
in  degrees 
Fahrenheit 

Pressure  In 
Atmospheres. 

Sulphurous  acid  

82° 

1-5 

46'4° 

2'5 

Cyanogen  gas  

82 

2'8 

Hvdiiodic  acid  

32 

4'0 

Ammoniacal  gas  

82 

4.4. 

50 

5 

Hydrochloric  acid  
Protoxide  of  azote  
Carbonic  acid  

82 

32 
32 

8-0 
87-0 
32-0 

51-8 
60 

43 
45 

Latent  heat  of  Evaporation. — It  has  already  been  stated,  that 
when  a  liquid  begins  to  boil,  the  subsequent  accessions  of  heat 

7* 


154  THEORY    OF   THE    STEAM-ENGINE. 

which  it  receives  go  not  to  increase  the  temperature,  but  to  ac- 
complish the  vaporisation.  The  heat  which  thus  ceases  to  be 
discoverable  by  the  thermometer  is  called  the  Latent  heat  of 
Vaporisation ;  and  experiments  have  shown,  that  if  the  heat 
thus  consumed  had  been  employed  to  raise  the  temperature  of 
the  water,  instead  of  boiling  it  away,  the  temperature  of  the 
water  would  have  been  raised  about  1,000  degrees  Fahrenheit, 
or  it  would  have  raised  about  1,000  times  the  same  weight  of 
water  that  is  boiled  off  1  degree  Fahrenheit. 

The  heat  consumed  in  evaporating  the  same  weight  of  dif- 
ferent liquids  varies  very  much,  but  it  does  not  follow  that  any 
of  them  would,  therefore,  be  better  than  water  as  an  agent  for 
the  generation  of  power,  as  the  bulk  of  the  resulting  vapour  in 
those  which  require  least  heat  is  small,  in  the  proportion  of  the 
smaller  quantity  of  heat  expended  hi  accomplishing  the  evapora- 
tion. Under  the  pressure  of  one  atmosphere,  or  14*7  Ibs.  per 
square  inch,  the  latent  heat  of  steam  from  water  has  been  found 
to  be  966'1.  Alcohol,  which  boils  at  172-2,  has  a  latent  heat  of 
evaporation  of  364*3.  Ether,  which  boils  at  95°,  has  a  latent 
heat  of  evaporation  of  162 '8°,  and  sulphuret  of  carbon,  which 
boils  at  114-8°,  has  a  latent  heat  of  evaporation  of  156°. 

The  most  important  of  the  researches  in  connection  with  this 
subject  are  those  which  have  reference  to  the  Latent  heat  of 
Steam,  and  this  topic  has  been  illustrated  by  the  researches  of 
various  experimentalists.  At  the  atmospheric  pressure,  and 
starting  at  the  temperature  of  212°,  the  following  estimates  of  the 
latent  heat  of  steam  have  been  formed  by  the  best  authorities  :— 


Watt 950-° 

Southern 945' 

Lavoisier 1000- 

Rumford. .  1008-8 


Despretz 955-8° 

Regnault 966'1 

Fabreand    )  9M.g 
Silbermann  ) 


The  experiments  which  are  generally  considered  to  be  the 
most  correct  in  connection  with  this  subject  are  those  of  M. 
Kegnault.  The  following  table,  taken  from  his  results,  show 
that  there  is  a  difference  of  about  150°  between  the  total  heat 
of  the  vapour  of  water  at  the  pressures  corresponding  to  32° 
and  446°  respectively . 


LATENT   HEAT   OF   STEAM. 


155 


SENSIBLE  AND  LATENT  HEAT   OE   STEAM.      BY  M.   KEGNAULT. 


Temperature 
In  degree* 

Fahrenheit. 

Latent  He»t 

Sum  of  Sensible 
and 
Latent  Heats. 

Temperature 
in  degrees 
Fahrenheit. 

Latent  Heat 

Sum  of  Sensible 
and 
Latent  Heats. 

32 

1092-6 

1124-6 

248 

936-6 

1187-6 

60 

1080-0 

1130-0 

266 

927-0 

1193-0 

68 

1067-4 

1135-4 

284 

914.4 

1198-4 

86 

1054-8 

1140-8 

302 

901-8 

1203-8 

104 

1042-2 

1146-2 

320 

889-2 

1209-2 

122 

1029-6 

1151-6 

838 

874-8 

1212-8 

140 

1017-0 

1157-0 

356 

862-2 

1218-2 

158 

1004-4 

1162-4 

374 

849-6 

1223-6 

176 

991-8 

1167-8 

392 

835-2 

1227-2 

194 

979-2 

1173-2 

410 

822-6 

1232-6 

212 

966-6 

1178-6 

428 

808-2 

1236-2 

230 

952-2 

1182-2 

446 

795-6 

1241-6 

Rules  for  connecting  the  temperature  and  elastic  force  of 
saturated  steam. — Various  formula  have  been  at  different  times 
propounded  for  deducing  the  elastic  force  of  saturated  steam 
from  its  temperature,  and  the  temperature  from  the  elastic  force. 
The  experiments  of  Mr.  Southern,  which  were  made  at  the  in- 
stance of  Boulton  and  "Watt,  led  to  the  adoption  of  the  follow- 
ing rules,  which,  though  not  quite  so  accurate  as  some  others 
which  have  since  been  arrived  at,  are  sufficiently  so  for  practical 
purposes,  and  being  intimately  identified  with  engineering  prac- 
tice, it  appears  desirable  to  retain  them. 

THE  TEMPEBATUEE  OF  8ATTJBATED  STEAM  BEING  GIVEN  IN  DEGEEE8 
FAHBENHEIT,  TO  FIND  THE  COEBESPONDING  ELASTIC  FOEOE  IN 
INCHES  OF  MEBCTJEY  BY  SOUTHEBN's  ETJLE. 

RULE. —  To  the  given  temperature  add  61 '3  degrees.  From  the 
logarithm  of  the  sum  subtract  the  logarithm  of  135*767, 
which  is  2-1327940.  Multiply  the  remainder  /by  5'13,  and  to 
the  natural  number  answering  to  the  sum,  add  the  constant 
fraction  •!.  The  sum  will  ~be  the  elastic  force  in  inches  of 
mercury. 

Example. — If  the  temperature  of  saturated  steam  be  250%3° 
Fahrenheit,  what  will  be  the  corresponding  elastic  force  in 
inches  of  mercury? 


156  THEORY   OF   THE    STEAM-ENGINE. 

Here  250-3  x  51-3  =  301-6     Log.    2.4794313 

135-767  Log.  2-1327940  subtract. 

remainder       0-3466373 
multiply  by  6 -13 


Natural  number  60-013  Log.  1-7782493 


This  natural  number  increased  by  -1  gives  us  60*113  inches 
of  mercury,  as  the  measure  of  the  elastic  force  sought. 


THE  ELASTIC  FORCE  OF  SATURATED  STEAM  BEING  GIVEN  IN  INCHES 
OF  MERCURY,  TO  FIND  THE  CORRESPONDING  TEMPERATURE  IN 
DEGREES  FAHRENHEIT  BY  SOUTHERN'S  RULE. 

EULE. — From  the  given  elastic  force  subtract  the  constant  frac- 
tion -1  ;  divide  the  logarithm  of  the  remainder  <by  5'13,  and 
to  the  quotient  add  the  logarithm  2'1327940.  Find  the 
natural  number  answering  to  the  sum  of  the  logarithms,  and 
from  the  number  thus  found  subtract  the  constant  51'3.  The 
remainder  will  le  the  temperature  sought  in  degrees  Fah- 
renheit. 

Example. — If  the  elastic  force  of  saturated  steam  balances  a 
vertical  column  of  mercury  238-4  inches  high,  what  is  the  tem- 
perature of  that  steam  ? 

Here  238-4  — 0-1  =  238-3 

Log.  238-3  =  2-3771240  -f-5'13  —  0-4633770 

2-1327940  add 


Natural  number  394-61  Log.  2-5961710 

Constant 51*3    subtract 


Required  temperature    343-31  degrees  Fahrenheit. 


The  temperature  of  the  steam  which  will  balance  such  a 
column  of  mercury,  has  been  ascertained  by  observation  to  be 
343-6  degrees. 


TEMPERATURE    AND    PRESSURE    OP    STEAM. 


157 


Experiments  have  been  made  by  the  French  Academy,  the 
Franklin  Institute  in  America,  and  various  other  experimental- 
ists, to  determine  the  elastic  force  of  steam  at  different  tempera- 
tures ;  but  of  all  these  experiments,  the  most  elaborate  and  the 
most  widely  accepted  are  those  of  M.  Eegnault.  The  results 
obtained  by  the  French  Academy  are  given  in  the  following 
table,  and  those  obtained  by  the  Franklin  Institute  are  very 
similar :  — 


PBESSTJBE   OF   STEAM  AT  DIFFERENT   TEMPEBATTJEES. 

Results  of  Experiments  made  by  the  French  Academy. 
An  atmosphere  is  reckoned  as  being  equal  to  29 '922  inches  of  mercury. 


Pressure  in 
Atmospheres. 

Temperature  in 
degrees  of 
Fahrenheit. 

Pressure  in 
Atmospheres. 

Temperature  in 
degrees  of 
Fahrenheit. 

1 

212° 

13 

386-66° 

H 

234 

14 

386-94 

2 

250-5 

15 

392-86 

tt 

263-8 

16 

398-48 

3 

275-2 

17 

403-83 

Si 

285 

18 

408-92 

4 

293-7 

19 

413-78 

*i 

300-3 

20 

418-46 

5 

307-5 

21 

422-96 

5i 

314-24 

22 

427-98 

6 

320-36 

23 

431-42 

U 

326-26 

24 

435-56 

7 

331-7 

25 

439-34 

n 

336-86 

30 

457-16 

8 

341-78 

35 

472-73 

9 

350-78 

40 

486-59 

10 

358-88 

45 

499-14 

11 

366-85 

50 

510-6 

12 

374 

Formula}  for  connecting  the  temperature  and  elastic  force  of 
steam  have  been  given  by  Young,  Tredgold,  Prony,  Biot,  Koche, 
Magnus,  Holtzmann,  Eankine,  Kegnault,  and  many  others — all 
more  or  less  complicated.  Eegnault  employs  different  formulae 


158  THEORY   Or   THE   STEAM-ENGINE. 

for  different  parts  of  the  thermometric  scale,  as  appears  from 
the  following  recapitulation  in  which  all  the  degrees  are  degrees 
centigrade : — 

EEGNATJLT'S  FORMULA.  FOE  THE  TEMPEEATUEE  AND  ELASTIC 
FOEOE  OF  STEAM. 

Between  0°  and  100°  the  formula  is 

Log.  F=a+6  a*! — c/3^, 

which  resembles  the  formula  previously  given  by  M.  Biot.  In 
this  formula  t  is  counted  from  0°  centigrade.  a=4'7384380 ;  Log. 

^=0-006865036;  Log.  j3± =1-9967249;  Log.  &=2'1340339,  and 
Log.  c=0-6116485. 

Between  100°  and  230°,  the  formula  he  used  is 

Log.  F=a — H>  ar — cj3r, 

in  which  r=£+20,  t  being  the  centigrade  temperature  reckoned 
fromO0.  Hence  0=6-2640348;  Log.  o=l'994049292 ;  Log. 
0=1-998848862 ;  Log.  5=0-1397743,  and  Log.  e=0-6924351. 

The  principal  properties  of  saturated  steam  as  deduced  from 
the  experiments  of  M.  Kegnault,  exhibiting  the  pressure,  the 
relative  volume,  the  temperature,  the  total  heat,  and  the  weight 
of  a  cubic  foot  of  steam  of  different  densities,  are  given  by  Mr. 
Clark  in  the  following  tables  : — 


REGNAULT'S  EXPERIMENTS  ON  STEAM. 


159 


PROPERTIES   OF   SATURATED    STEAM. 
BY    M.  BEGNATJXT. 


Total  Pressure  per 
Square  Inch. 

Eelatlve  Volume  . 

Temperature. 

Total  Heat. 

Weight  of  one  Cubic 
Foot. 

|  Total  Pressure  per 
|  Square  Inch. 

Relative  Volume. 

Temperature. 

Total  Heat. 

Weight  of  One  Cubic 
Foot. 

Us. 

Fahr. 

Fahr. 

Lbs. 

Lbs. 

Fahr. 

Fahr. 

Lbt. 

15 

1669 

213-1 

1178-9 

•0373 

48 

573 

278-4 

1198-8 

•1087 

16 

1572 

216-3 

1179-9 

•0397 

49 

562 

279-7 

1199-2 

•1108 

17 

1487 

219-5 

1180-9 

•0419 

50 

552 

281-0 

1199-6 

•1129 

18 

1410 

222-5 

1181-8 

•0442 

51 

542 

282-3 

1200-0 

•1150 

19 

1342 

225-4 

1182-7 

•0465 

52 

532 

283-5 

1200-4 

•1171 

20 

1280 

228-0 

1183-5 

•0487 

63 

523 

284-7 

1200-8 

•1192 

21 

1224 

230-6 

1184-3 

•0510 

54 

514 

285-9 

1201-1 

•1212 

22 

1172 

233-1 

1185-0 

•0532 

55 

506 

287-1 

1201-5 

•1232 

23 

1125 

235-5 

1185-7 

•0554 

56 

498 

288-2 

1201-8 

•1252 

24 

1082 

237-9 

1186-5 

•0576 

57 

490 

289-3 

1202-2 

•1272 

25 

1042 

240-2 

1187-2 

•0598 

58 

482 

290-4 

1202-5 

•1292 

26 

1005 

242-3 

1187-9 

•0620 

59 

474 

391-6 

1202-9 

•1314 

27 

971 

244-4 

1188-5 

•0642 

60 

467 

292-7 

1203-2 

•1335 

28 

939 

246-4 

1189-1 

•0664 

61 

460 

293-8 

1203-6 

•1356 

29 

909 

248-4 

1189-7 

•0686 

62 

453 

294-8 

1203-9 

•1376 

30 

881 

250-4 

1190-3 

•0707 

63 

447 

295-9 

1204.2 

•1396 

31 

855 

252-2 

1190-8 

•0729 

64 

440 

296-9 

1204-5 

•1416 

82 

830 

254-1 

1191-4 

•0751 

65 

434 

298-0 

1204-8 

•1436 

33 

807 

255-9 

1192-0 

•0772 

66 

428 

299-0 

1205-1 

•1456 

34 

785 

257-6 

1192-5 

•0794 

67 

422 

300-0 

1205-4 

•1477 

35 

765 

259-3 

1193-0 

•0815 

68 

417 

300-9 

1205-7 

•1497 

36 

745 

260-9 

1193-5 

•0837 

69 

411 

301-9 

1206-0 

•1516 

37 

727 

262-6 

1194-0 

•0858 

70 

406 

302-9 

1206-3 

•1535 

38 

709 

264-2 

1194-5 

•0879 

71 

401 

303-9 

1206-6 

•1555 

39 

693 

265-8 

1195-0 

•0900 

72 

396 

304-8 

1206-9 

•1574 

40 

677 

267-3 

1195-4 

•0921 

73 

391 

305-7 

1207-2 

•1595 

41 

661 

268-7 

1195-9 

•0942 

74 

386 

306-6 

1207-5 

•1616 

42 

647 

270-2 

1196-3 

•0963 

75 

381 

307-5 

1207-8 

•1636 

43 

634 

271-6 

1196-8 

•0983 

76 

377 

308-4 

1208-0 

•1656 

44 

621 

273-0 

1197-2 

•1004 

77 

372 

309-3 

1208-3 

•1675 

45 

608 

274-4 

1197-6 

•1025 

78 

368 

310-2 

1208-6 

•1696 

46 

595 

275-8 

1198-0 

•1046 

79 

364 

311-1 

1208-9 

•1716 

47 

584 

277-1 

1198-4 

•1067 

80 

359 

312-0 

1209-1 

•1736 

160 


THEORY   OF   THE    STEAM-ENGINE. 


PROPERTIES   OF  SATURATED  STEAM — Continued. 
BY    M.  BEGNAULT. 


Total  Pressure  per 
Square  Inch. 

Kelative  Volume. 

Temperature. 

Total  Heat. 

Weight  of  one  Cubic 
Foot. 

|  Total  Pressure  per 
Square  Inch. 

Relative  Volume. 

Temperature. 

Total  Heat. 

Weight  of  One  Cubic 
Foot. 

Lb« 

Fahr. 

Falvr. 

Lbs. 

Lbs. 

Fahr. 

Fahr. 

Lbs. 

81 

355 

312-8 

1209-4 

•1756 

114 

261 

337-4 

1216-8 

•2388 

82 

351 

313.6 

1209-7 

•1776 

115 

259 

338-0 

1217-0 

•2406 

83 

348 

314-5 

1209-9 

•1795 

116 

257 

338-6 

1217-2 

•2426 

84 

344 

315-3 

1210-1 

•1814 

117 

255 

339-3 

1217-4 

•2446 

85 

340 

316-1 

1210-4 

•1833 

118 

253 

339-9 

1217-6 

•2465 

86 

337 

316-9 

1210-7 

•1852 

119 

251 

340-5 

1217-8 

•2484 

87 

333 

317-8 

1210-9 

•1871 

120 

249 

341-1 

1218-0 

•2503 

88 

330 

318-6 

1211-1 

•1891 

121 

247 

341-8 

1218-2 

•2524 

89 

326 

319-4 

1211-4 

•1910 

122 

245 

342-4 

1218-4 

•2545 

90 

323 

320-2 

1211-6 

•1929 

123 

243 

343-0 

1218-6 

•2566 

91 

320 

321-0 

1211-8 

•1950 

124 

241 

343-6 

1218-7 

•2587 

92 

317 

321.7 

1212-0 

•1970 

125 

239 

344-2 

1218-9 

•2608 

93 

313 

322-5 

1212-3 

•1990 

126 

238 

344-8 

1219-1 

•2626 

94 

310 

323-3 

1212-5 

•2010 

127 

236 

345-4 

1219-3 

•2644 

95 

307 

324-1 

1212-8 

•2030 

128 

234 

346-0 

1219-4 

•2662 

96 

305 

324-8 

1213-0 

•2050 

129 

232 

346-6 

1219-6 

•2680 

97 

302 

325-6 

1213-3 

•2070 

130 

231 

347-2 

1219-8 

•2698 

98 

299 

326-3 

1213-5 

•2089 

132 

228 

348-3 

1220-2 

•2735 

99 

296 

327-1 

1213-7 

•2108 

134 

225 

349-5 

1220-6 

•2771 

100 

293 

327-8 

1213-9 

•2127 

136 

222 

350-6 

1220-9 

•2807 

101 

290 

328-5 

1214-2 

•2149 

138 

219 

351-8 

1221-2 

•2846 

102 

288 

329-1 

1214-4 

•2167 

140 

216 

352-9 

1221-5 

•2885 

103 

285 

329-9 

1214-6 

•2184 

142 

213 

354-0 

1221-9 

•2922 

104 

283 

330-6 

1214-8 

•2201 

144 

210 

355-0 

1222-2 

•2959 

105 

281 

331-8 

1215-0 

•2218 

146 

208 

356-1 

1222-6 

•2996 

106 

278 

331-9 

1215-2 

•2230 

148 

205 

357-2 

1222-9 

•3033 

107 

276 

332-6 

1215-4 

•2258 

150 

203 

358-3 

1223-2 

•3070 

108 

273 

333-3 

1215-6 

•2278 

160 

191 

363-4 

1224-8 

•3263 

109 

271 

334-0 

1215-8 

•2298 

170 

181 

368-2 

1225-1 

•3443 

110 

269 

334-6 

1216-0 

•2317 

180 

172 

372-9 

1227-7 

•3623 

111 

267 

335-3 

1216-2 

•2334 

190 

164 

377-5 

1229-1 

•3800 

112 

265 

336-0 

1216-4 

•2351 

200 

157 

381-7 

1230-3 

•3970 

113 

263 

336-7 

1216-6 

•2370 

KEGNAULT'S  EXPERIMENTS  ON  VAPOURS. 


161 


M.  Regnault  extended  his  researches  to  the  pressure  of  other 
vapours,  beside  that  of  water.  The  following  are  the  results 
he  obtained  with  alcohol,  ether,  sulphuret  of  carbon,  chloroform, 
and  essence  of  turpentine : 


TEMPEBATUBE   AND   ELASTIC   FOBCE   OF   THE  VAPOTJE3   OF   DIFFEB- 

ENT  LIQUIDS.      BY   M.   BEGNAULT. 
[A  millimetre  is  one  thousandth  part  of  a  metre,  or  0-03937  of  an  inch.] 


Tensionoftho  Va- 
pour of  Alcohol. 

Tension  of  the  Va- 
pour of  .Ether. 

Tension  of  the  Va- 
pour of  Sulphuret 
of  Carbon. 

Tension  of  Vapour 
of  Chloroform 
by  Tension  in 
Vacua. 

Tension  of  the  Va- 
pour of  Essence 
of  Turpentine. 

Temperature  in  T)e- 
gNM  Ceutrlgrade. 

1  & 

II 

£S 

P 

1 

~   9 

go 

2fr 

—   9 

SS 

.2*3 
£•5 
s' 

I* 

ft 

*j 

*i 
ll 

t* 
ll 

IB 

H 

if 
§1 

a  3 

£•3 

|! 

P^ 
—•9 
|| 
P 

|| 

|6 

3f? 
11 

£% 
£•3 

i! 

&4 

a| 

i 

H 

1* 

33 

;s§ 

£% 
£"3 

i1 

—21° 
—20 
—10 
0 
10 
20 
80 
40 
50 
60 
70 
80 
90 
100 
110 
120 
180 
140 
150 
152 

8-12 
3-34 
6-50 
12-78 
24-08 
44-0 
78-4 
184-1 
220-8 
860-0 
689-2 
812-S 
1190-4 
1685-0 
2351-8 
8207-8 
4331-2 
5687-7 
7257-8 
7617-8 

—20 
—10 
0 
10 
20 
80 
40 
60 
60 
70 
80 
90 
100 
101 

69-2 
113-2 
182-8 
286-5 
434-8 
687-0 
918-6 
1268-0 
1730-8 
2309-5 
2947-2 
8899-0 
4920-4 
7076-2 

—16° 
—10 
0 
10 
20 
80 
40 
60 
60 
70 
90 
80 
100 
110 
120 
180 
136 

58-8 
79-0 
127-8 
199-3 
298-2 
434-6 
617-5 
862-7 
1162-6 
1549-0 
2080-5 
2623-1 
8321-3 
4186-3 
6121-6 
6260-6 
7029-2 

+  10° 
20 
80 
36 

130-4 
190-2 
276.1 
842.2 

0° 

10 
20 
80 
40 
50 
60 
70 
80 
90 
100 
110 
120 
180 
140 
160 
160 
170 
180 
190 
200 
210 
220 
222 

2-1 
2-8 
4-8 
7-0 
11-2 
17-2 
26-9 
419 
61-2 
91-0 
184-9 
187-8 
267-0 
847-0 
462-8 
604-5 
777-2 
989-0 
1225-0 
1514-7 
1865-6 
2251-2 
2690-8 
2778-5 

by  the  method 
of  ebullition. 

'86* 
40 
60 
60 
70 
80 
90 
100 
110 
120 
130 

813-4 
864-0 
524-8 
788-0 
976-2 
1367-8 
1811-6 
2354-6 
8020-4 
8818-0 
4721-0 

Unit  of  heat. — It  is  convenient  with  the  view  of  enabling  us 
to  compare  the  quantities  of  heat  in  different  bodies  to  fix  upon 
some  thermal  unit,  by  which  quantities  of  heat  may  be  measur- 
ed ;  and  the  thermal  unit  employed  in  this  country  is  the  quan- 


162  THEORY   OF   THE    STEAM-ENGINE. 

tity  of  heat  which  is  required  to  raise  a  pound  of  pure  water  at 
its  point  to  maximum  density,  through  one  degree  Fahrenheit. 
In  France  the  thermal  unit  employed  is  the  quantity  of  heat  re- 
quired to  raise  a  kilogramme  of  pure  water  at  its  point  of  great- 
est density  through  one  degree  Centigrade.  A  kilogramme  is 
2'20462  Ibs.  avoirdupois,  or  a  pound  avoirdupois  is  0-453593  of 
a  kilogramme.  A  degree  Centigrade  is  1'8  degrees  Fahrenheit ; 
and  a  degree  Fahrenheit  is  0'555  of  a  degree  Centigrade.  There 
are  3'96832  British  thermal  units  in  a  French  thermal  unit, 
and  there  is  0-251996  of  a  French  thermal  unit  in  a  British 
thermal  unit. 


SPECIFIC   HEAT. 

The  specific  heat  of  a  substance  is  an  expression  for  the  quan- 
tity of  heat  in  any  given  weight  of  it  at  a  certain  temperature, 
just  as  its  specific  gravity  is  an  expression  for  the  quantity  of 
matter  in  a  given  bulk.  Specific  heat  is  most  conveniently  ex- 
pressed by  a  reference  to  the  number  of  thermal  units  consumed 
in  producing  a  given  elevation  of  temperature  in  the  body  under 
consideration ;  or,  if  the  weight  of  a  heated  body  immersed  in 
water  be  multiplied  by  the  temperature  it  loses,  and  the  weight 
of  water  be  multiplied  by  the  temperature  it  gains,  the  quotient 
obtained  by  dividing  the  latter  product  by  the  former,  will  be  the 
specific  heat  of  the  body.  The  specific  heats  of  various  substan- 
ces have  been  experimentally  ascertained  and  recorded  in  tables, 
in  which  the  specific  heat  of  water  is  reckoned  as  unity.  Thus, 
the  specific  heat  of  air  is  -2379,  or  it  is  4*207  times  less  than  that 
of  water.  An  amount  of  heat,  therefore,  which  would  raise  a 
pound  of  water  1  degree,  would  raise  a  pound  of  air  4'207 
degrees. 

The  following  tables  of  specific  heats  are  derived  from  the 
experiments  of  the  best  authorities,  and  chiefly  from  those  of 
M.  Eegnault.  The  specific  heat  of  ice  is  given  on  the  authority 
of  M.  Person. 


SPECIFIC   HEATS   OF   DIFFEREKT   SUBSTANCES. 


163 


SPECIFIC   HEATS   OF   SOLIDS. 

The  specific  heat  of  water  being  reckoned  as  unity. 


NAME  OP  SUBSTANCE. 

Specific 
Heat. 

NAJtE  OF  SUBSTANCE. 

Specific 
Heat. 

Iron  

0-11379 

Gold  

0-03244 

Cast-iron  (white)  

0-12983 

Platinum  

0-03243 

Steel,  soft  

0-11650 

Glass  

0-19768 

"     tempered  .  .    .  . 

0-11750 

Sulphur  

0-20259 

Copper  

0-09515 

Silicia  

0-19132 

Brass  

0-09391 

Carbon  

0-24111 

Zinc  

0-09555 

Coke  

0-20200 

Lead  

0-03140 

Diamond  

0-14687 

Tin  

0-05623 

Phosphorus  

0-18870 

Silver  

0-05701   ' 

Ice  

0-50400 

SPECIFIC   HEATS   OF   GASES   AND   YAPOTJKS. 

The  specific  heat  of  water  leing  reckoned,  as  unity. 


Specifi 

:  Heat. 

For  equal 
Weights. 

For  equal 
Volumes. 

Oxygen  .  . 

0-2182 

0-2412 

1-1056 

Nitrogen  

0-2440 

0-2370 

0-9713 

Hydrogen  

3-4046 

0-2356 

0-0692 

Chlorine  

0-1214 

0-2962 

2-4400 

Protoxide  of  nitrogen  

0-2238 

0-3413 

1-5250 

Binoxide  of  nitrogen  

0-2315 

0-2406 

1-0390 

Carbonic  oxide  

0-2479 

0-2399 

0-9674 

Carbonic  acid  

0-2164 

0-3308 

1-5290 

Sulphuret  of  carbon  

0-1576 

0-4146 

2-6325 

Sulphurous  acid  

G'1553 

0-3489 

2-2470 

Ammonia.  

0-5080 

0-2994 

0-5894 

Protocarburet  of  hydrogen  — 

0-5929 

0-32-77 

0-6527 

Bi-carburet  of  hydrogen  
Water  vapour,  or  steam  
Alcohol  vapour  

0-3694 
0-4750 
0-4513 

0-3572 
0-2950 
0-7171 

0-9672 
0-6210 
1-5890 

0-4810 

1-2296 

2-5563 

Chloroform  vapour  

0-1568 

0-8310 

5-3000 

Turpentine  vapour  

0-5061 

2-3776 

4-6978 

164 


THEORY   OF   THE    STEAM-ENGINE. 


SPECIFIC   HEATS   OF  LIQUIDS. 

The  specific  heat  of  water  heing  reckoned  as  unity. 


NAME  OF  LIQUID. 

Specific 
Heat. 

NAME   OF   LIQUID. 

Specific 
Heat. 

Mercury  

0-0333 

Petroleum  

0-4684 

Turpentine  

0-4672   i 

Solution  Clilo.  Lime  .  . 

0-6448 

Gin  

0-4770 

Spirit  of  Wine  at  97.. 

0-6588 

Olive  Oil  

0-3096 

Acetic  Acid  

0-6501 

It  will  be  observed  from  the  foregoing  tables  that  the  specific 
heat  of  steam  is  nearly  the  same  as  the  specific  heat  of  ice.  The 
specific  heat  of  water,  and  also  of  air,  occupying  the  same  volume, 
is  found  to  be  the  same  at  all  temperatures  between  boiling  and 
freezing,  and  the  specific  heat  of  air  under  a  constant  pressure 
may  be  taken  at  0-2379.  In  other  words,  it  requires  just  the 
same  amount  of  heat  to  raise  water  and  air  one  degree  in  tem- 
perature at  any  one  part  of  the  thermometric  scale  as  at  any 
other ;  and  the  heat  required  to  heat  a  pound  of  air  1  degree  is 
only  -2379,  or  less  than  one-fourth  of  the  quantity  required  to 
heat  a  pound  of  water  one  degree.  If  therefore  a  pound  of  wa- 
ter at  60°  has  transferred  to  it  the  heat  in  a  pound  of  air  at 
1000°,  the  water  will  not  acquire  as  much  elevation  of  tempera- 
ture as  the  air  loses,  but  only  -2379  of  that  temperature. 


EATIO  OF  SPECIFIC   HEATS    OF   GASES     TINDER   CONSTANT   PEESStTEB 
TO   THE   SPECIFIC   HEATS   UNDER   CONSTANT   YOLUME. 

When  air  is  compressed  it  generates  heat,  as  is  shown  in 
the  syringe  in  which  a  piece  of  tinder  is  lighted  by  the  heat  pro- 
duced by  the  sudden  compression  of  air;  and,  contrariwise, 
when  air  or  any  other  gas  is  expanded  it  produces  cold.  When, 
therefore,  a  cubic  foot  of  air  of  the  atmospheric  pressure  is  heat- 
ed until  its  pressure  is  doubled,  it  will  have  a  certain  tempera- 
ture which  will  fall  if  the  air  is  suffered  to  expand  into  a  volume 
of  two  cubic  feet,  and  to  restore  the  previous  temperature  more 
heat  must  be  added.  It  will  take  more  heat,  therefore,  to  heat 


SPECIFIC   HEATS  AND    SPECIFIC   GRAVITIES. 


165 


a  cubic  foot  of  air  to  a  given  temperature,  if  it  be  suffered  to  ex- 
pand, than  if  it  be  not  suffered  to  expand ;  and  only  that  part 
of  the  heat  is,  properly  speaking,  specific  heat,  which  is  shown 
by  the  rise  of  temperature,  that  which  is  absorbed  in  enlarging 
the  volume  being,  in  point  of  fact,  latent  heat.  Both  kinds  of' 
heat,  however,  are  very  generally  called  specific  heat,  but  as  the 
quantities  are  very  different,  it  follows  that  there  are  two  kinds 
of  specific  heat — the  one  the  specific  heat  under  a  constant  vol- 
ume, and  the  other  the  specific  heat  under  the  increased  volume 
to  which  the  body  naturally  enlarges.  It  is  only  in  the  case  of 
gases  that  there  is  a  material  difference  between  these  specific 
heats.  But  in  the  case  of  gases  the  difference  is  very  consider- 
able, and  it  is  found  that  the  specific  heat  under  a  constant  press- 
ure divided  by  the  specific  heat  under  a  constant  volume,  is 
equal,  in  the  case  of  air,  to  1-408;  or,  in  other  words,  the  speci- 
fic heat  of  air  under  a  constant  pressure  is  1'408  times  greater 
than  that  of  air  under  a  constant  volume.  The  specific  heat  of 
air  under  a  constant  pressure,  may  be  taken  at  '2379,  which 
makes  the  specific  heat  under  a  constant  volume  '169.  The  fol- 
lowing tables  of  the  specific  heats,  and  some  other  properties  of 
solids,  liquids,  and  gases  are  given  by  Mr.  Bankine : — 


SPECIFIC  HEATS  AND  SPECIFIC   GRAVITIES  OF  METALS. 


Name  of  Metal. 

Weight  of  a 
cubic  foot  In  Ib*. 
Do. 

Specific 
Gravity. 
S.G. 

Expansion 
from 
32°  to  212°. 

E 

Specific 
Heat 
C. 

Specific  Heat 

in  foot-pound*. 

H 

Brass  

487  to  533 

7'8  to  8-5 

•00216 

Bronze  

524 

8-4 

•00181 

Copper.  

537  to  556 

8-6  to  8-9 

•00184 

•0951 

73-8 

Gold.  

11S6  to  1224 

19-  to  19-6 

•0015 

•0298 

28  '0 

Iron,  cast  
Iron,  wrought.. 
Lead  

444 

480 
712 

7-11 
7-69 
11-4 

•0011 
•0012 
•0029 

•1188 
•0298 

87-8 
22-6 

Platinum  
Silver  

1811  to  1373 
655 

21  to  22 
10-5 

•0009 
•002 

•0814 
•0557 

24-2 
43-0 

fiteel  

490 

7-86 

•0012 

Tin  

462 

7-4 

•0022 

•0514 

89-7 

Zinc       

436 

7^2 

00294 

•0927 

71  '6 

Ice  

676 

0-92 

•604 

889 

166 


THEORY   OF   THE    STEAM-ENGINE. 


SPECIFIC   HEATS   AND   SPECIFIC    GEAVITIES   OF   LIQUIDS. 


Name  of  Liquid. 

Do. 

S.  G. 

E. 

c. 

K. 

Water,  pure  at  39-1  °  

62-425 

1-000 

0-04775 

1-000 

772-0 

"      sea,  ordinary  

64-05 

1-026 

0-05 

Alcohol,  pure  

49-38 

0-791 

0-1112 

"         proof  spirit  
^Ether  

57-18 
44-70 

0-916 
0-716 

0-517 

399-1 

Mercury  

848-75 

13-596 

0-018153 

0-033 

25-5 

Naphtha  

52-94 

0-848 

Oil,  Linseed  

68-68 

0-940 

0-08 

"     Olive  

57-12 

0-915 

0-08 

"     Whale  

57-62 

0-923 

"     of  Turpentine  

54-31 

8-870 

0-07 

Petroleum  

54-81 

0-878 

DENSITIES,   VOLUMES,   BATES   OF   EXPANSION,    AND   SPECIFIC 
HEATS   OF   GASES. 


Name  of  Gas. 

Weight 

cubic 
foot  in 
Iba. 
De. 

Volume 
in  cubic 
feet  of 
lib. 
V... 

Expan- 
sion 
from 
32"  to 
21J° 
E. 

Specific  Heat  in 
degrees  Fahr. 

Specific  Heat  in 
foot-pounds. 

Under 

volume. 
Cv. 

Under 
constant 
pressure 
Cp. 

Under 
constant 
volume. 
Kv. 

Under 
constant 
pressure. 
Kp. 

Air  

0-080728 
0-089256 
0-005592 
0-05022* 
0-2093* 

0-2137* 

0-12259* 
0-12344 
0-0795 
0-078411 

0-563* 

12-387 
11-204 
178-83 
19-913* 
4-777* 

4-679* 

8-157* 
8-101 
12-58 
12-753 

1-7762* 

•365 
•867 
•366 
•365* 

•365* 
•370 

0-169 
0-156 
2-410 
0-365* 

0-173 

0-238 
0-218 
3-405 
0-475 
0-481 

0-1575 

0-217 
0-869 
0-244 

180-3 
120-2 
1860-6 
281-3* 

138-6 

188-45 
168-3 
2628-7 
866-7 
871-3 

121-6 

167-0 
284-9 

188-4 

Hydrogen  

.(Ether  vapour.. 
Bisulph.  carbon 
vapour  
Carbonic     acid 
(ideal)  

Ditto  (actual).. 
Olefiant  gas.  .  .  . 
Nitrogen  
Vapour  of  Mer- 
cury   

An  asterisk  (*)  is  affixed  to  the  results  computed  for  the  ideal  condition  of  a  perfect  gas. 

In  these  tables  the  volumes  are  taken  at  the  temperature  of 
melting  ice,  or  32° ;  except  in  the  case  of  water,  which  is  taken 
at  the  temperature  of  maximum  density,  or  39-1°.  The  pressure 
is  taken  at  the  usual  atmospheric  pressure  of  2116-4  Ibs.  upon 
the  square  foot. 

D0  is  the  density  or  weight  of  1  cubic  foot  of  the  substance  in 


SPECIFIC   HEATS   IN   FOOT-POUNDS.  167 

Ibs.  avoirdupois  under  the  pressure  of  one  atmosphere,  or  2116-4 
Ibs.  on  the  square  foot. 

V0  is  the  volume  in  cubic  feet  of  1  pound  avoirdupois  of  the 
substance  at  the  foregoing  temperature  and  pressure.  S.G.  is 
the  specific  gravity,  water  being  taken  as  unity.  E  is  the  expan- 
sion of  unity  of  volume  for  fluids,  and  unity  of  length  for  solids, 
at  the  temperature  of  melting  ice,  in  being  raised  from  the  tem- 
perature of  melting  ice  to  the  temperature  of  boiling  water  un- 
der the  pressure  of  one  atmosphere.  C  is  the  specific  heat  in  de- 
grees Fahrenheit,  the  specific  heat  of  water  being  reckoned  as 
unity,  and  Cy  is  the  specific  heat  under  a  constant  volume,  while 
Op  is  the  specific  heat  under  a  constant  pressure.  K  is  the  speci- 
fic heat,  reckoned  not  in  degrees  of  temperature,  but  in  the  equiv- 
alent value  of  pounds  raised  1  foot  high.  It  has  already  been 
explained  that  there  is  as  much  power  in  the  form  of  heat  ex- 
pended in  raising  a  pound  of  water  1  degree  in  temperature  as 
would  raise  772  Ibs.  to  the  height  of  1  foot ;  and  772  foot-pounds 
is,  consequently,  the  mechanical  equivalent  of  a  pound  of  water 
raised  1  degree.  Now  as  the  specific  heats  of  all  bodies  are  de- 
tenninable  by  the  temperature  to  which  a  pound  of  the  sub- 
stance will  raise  a  pound  of  water,  and  as  the  accession  of  heat 
which  a  pound  of  water  receives  is  transformable  into  its  equiv- 
alent amount  of  mechanical  power,  it  follows  that  the  specific 
heats  of  all  bodies  may  be  represented  by  the  amount  of  me- 
chanical power  in  foot-pounds,  which  is  the  equivalent  of  the 
heat  consumed  in  raising  a  pound  of  any  of  these  bodies  through 
one  degree  of  temperature.  Such  specific  heats,  accordingly,  are 
those  represented  in  the  tables  by  the  letter  K ;  the  expression 
KT  being  the  specific  heat  in  foot-pounds  of  unity  of  weight  under 
a  constant  volume,  and  Kp  the  specific  heat  of  the  same  weight 
under  a  constant  pressure.  The  value  of  Zp  -4-  KT,  Mr.  Kankine 
states,  is  in  the  case  of  air,  1-408  ;  oxygen,  1-4;  hydrogen,  1-413 ; 
nitrogen,  1-409;  and  steam,  considered  as  a  perfect  gas,  1-304; 
or,  in  other  words,  the  specific  heat  tinder  a  constant  volume  is 
to  the  specific  heat  under  a  constant  pressure  as  1  to  1  '4  in 
the  case  of  oxygen,  differing  slightly  in  the  case  of  the  other 
gases. 


168  THEORY   OF   THE   STEAM-ENGINE. 

PHENOMENA  OF  EBULLITION. 

Influence  of  Viscosity  or  Molecular  Attraction. — Salts  dis- 
solved in  water  will  raise  the  temperature  of  its  boiling-point. 
The  attraction  of  a  salt  for  water  being  greater  than  the  attrac- 
tion of  the  particles  of  the  water  for  one  another,  will  resist  the 
repellent  force  of  the  heat  to  some  extent.  Mechanical  pressure 
applied  to  the  water  has  the  same  operation.  Hence,  water 
boils  in  a  vacuum  at  a  lower  temperature  than  under  the 
pressure  of  the  atmosphere,  and  it  also  boils  at  a  lower  tem- 
perature under  the  pressure  of  one  atmosphere  than  under  a 
pressure  of  several  atmospheres.  Water,  which  has  been  well 
purged  of  air  by  boiling,  does  not  pass  into  the  state  of  steam 
when  heated  in  clean  glass  vessels,  until  it  has  attained  a  tem- 
perature considerably  higher  than  its  ordinary  boiling  point; 
and  when  the  steam  finally  forms,  it  forms  rather  by  a  jumping 
motion,  or  by  a  sudden  shock,  than  by  a  gradual  and  silent  dis- 
engagement. M.  Magnus  found  that  water  well  cleared  of  air 
may  be  raised  to  a  temperature  of  105°  or  106°  Centigrade 
before  boiling,  if  the  glass  vessel  in  which  it  was  heated  were 
perfectly  clean ;  but  if  the  vessel  were  soiled,  or  if  dust  or 
other  foreign  particles  were  suffered  to  enter  it,  the  temperature 
would  fall  to  the  usual  boiling  point  of  100°  Centigrade.  The 
sides  of  metallic  vessels,  or  sawdust,  metal  filings,  or  insoluble 
particles  of  almost  any  kind,  introduced  into  a  liquid,  lower  its 
boiling-point.  These  particles  are  not  at  every  point  completely 
moistened  by  the  water,  and  they  have  a  less  attraction  for  the 
particles  of  the  fluid  than  the  particles  of  the  fluid  have  for  one 
another.  In  the  process  of  ebullition,  therefore,  the  steam 
chiefly  forms  around  those  particles  and  seems  to  come  out  of 
them,  and  the  boiling-point  is  lowered  by  the  greater  facility 
they  occasion  to  the  disengagement  of  the  steam.  M.  Donny, 
by  freeing  water  carefully  from  air,  succeeded  in  raising  it  to  a 
temperature  of  135°  without  boiling ;  but  at  this  temperature 
steam  was  suddenly  formed,  and  a  portion  of  the  water  was 
projected  forcibly  from  the  tube.  M.  Donny  concludes,  from 
his  experiments,  that  the  mutual  force  of  cohesion  of  the  parti- 


SPHEROIDAL   CONDITION   OF   LIQUIDS.  169 

cles  of  water  is  equal  to  a  pressure  of  about  three  atmospheres, 
and  to  this  strong  cohesive  force  he  attributes  the  irregular 
jumping  motion  observed  in  ebullition,  and  also  some  of  those 
explosions  of  steam-boilers  which  heretofore  have  perplexed  en- 
gineers. It  is  well  known  that  cases  have  occurred  in  which  an 
open  pan  of  boiling  water  has  exploded,  producing  fatal  results, 
and  such  explosions  cannot  be  explained  on  the  usual  hypothesis. 
M.  Donny  says  that  liquids  by  boiling  lose  the  greater  part  of  the 
air  which  they  hold  in  solution,  and  therefore  the  molecular  at- 
traction begins  to  manifest  itself  in  a  sensible  manner.  The 
liquid  consequently  attains  a  temperature  considerably  above 
its  normal  boiling-point,  which  determines  the  appearance  of 
new  air-bubbles,  when  the  liquid  separates  abruptly,  a  quantity 
of  vapour  forms,  and  the  equilibrium  is  for  the  moment  restored. 
The  phenomenon  then  recurs  again  with  increased  violence,  and 
an  explosion  may  eventually  ensue. 

Spheroidal  Condition  of  Liquids  on  Hot  Surfaces. — If  a  drop 
of  water  or  other  liquid  be  thrown  upon  a  hot  metal  plate  or 
other  highly  heated  surface,  it  does  not  moisten  the  surface  or 
diffuse  itself  over  it,  but  forms  a  flattened  ellipsoidal  mass ;  and 
if  the  drop  be  sufficiently  small,  it  forms  a  minute  spheroid, 
which  revolves  rapidly  round  a  shifting  axis,  and  evaporates 
very  slowly  without  entering  the  state  of  ebullition.  From 
Church's  experiments  it  appears  that  it  is  necessary  for  the 
liquid  to  emit  vapour  before  it  can  assume  the  spheroidal  state. 
Molten  lead  dropped  upon  a  very  hot  platinum  plate  did  not  as- 
sume the  spheroidal  state,  whereas  mercury  dropped  upon  this 
plate  assumed  the  spheroidal  state  at  once.  The  most  remark- 
able experiments,  however,  which  have  been  made  in  illustration 
of  the  phenomena  of  the  spheroidal  state  are  those  of  M.  Bou- 
tigny,  and  to  him  engineers  are  mainly  indebted  for  calling  their 
attention  to  the  subject.  One  of  the  most  singular  results  ob- 
tained by  M.  Boutigny  is  the  power  of  making  ice  in  a  red  hot 
crocible.  A  small  crucible  or  capsule  of  platinum  being  made 
white  hot,  some  anhydrous  sulphurous  acid  in  the  liquid  state  is 
poured  into  it.  The  boiling-point  of  this  liquid  is  as  low  as  14° 
Fahrenheit ;  but  as  it  immediately  on  being  projected  into  the 
8 


170  THEORY   OF   THE    STEAM-ENGINE. 

capsule  assumes  the  spheroidal  state,  it  remains  upon  the  white 
hot  metal  without  touching  it ;  and  if  a  few  drops  of  water  be 
now  let  fall  upon  the  liquid  acid,  the  water  will  be  immediately 
frozen,  and  a  piece  of  ice  may  be  turned  out  of  the  crucible. 
M.  Boutigny  has  also  shown  that  if  acids  and  alkalies  in  solution 
be  poured  into  a  clean  red  hot  platinum  crucible  they  will  not 
unite,  but  both  will  assume  the  spheroidal  state  and  roll  about 
the  bottom  of  the  crucible  without  entering  into  combination. 
Not  merely  the  gravitation  of  the  liquid,  therefore,  but  also  its 
chemical  affinity,  appears  to  be  superseded  by  the  causes  which 
make  it  assume  the  spheroidal  state. 

When  a  liquid  assumes  the  spheroidal  state  it  does  not  wet 
the  surface,  but  appears  to  avoid  touching  it,  like  water  sprinkled 
upon  grease.  Instead  of  entering  into  violent  ebullition  when 
it  reaches  the  hot  surface,  its  temperature  will  rise  very  little, 
and  the  drops  of  liquid  will  either  remain  at  rest  or  will  acquire 
a  gyratory  motion.  When  the  surface  is  cooled  down  to  400° 
to  500°,  depending  on  the  nature  of  the  surface  and  also  on  the 
nature  of  the  liquid,  the  liquid  will  begin  to  diffuse  itself,  and 
will  be  suddenly  scattered  in  all  directions.  The  requisite  tem- 
perature of  a  platinum  plate  to  make  water  at  the  boiling-point 
assume  the  spheroidal  state  is  120°  Centigrade,  or  248°  Fahren- 
heit ;  but  if  glass  be  used  instead  of  platinum,  the  temperature 
must  be  raised  to  180°  Centigrade,  or  324°  Fahrenheit.  For 
water  at  0°  Centigrade,  the  temperatures  required  are  400°  and 
800°  respectively. 

When  water  assumes  the  spheroidal  state,  it  is  possible  by 
placing  the  eye  on  the  level  of  the  hot  surface  to  see  between 
the  surface  and  the  liquid.  The  electric  circuit,  moreover,  is  in- 
terrupted, showing  that  there  is  no  actual  contact  between  the 
liquid  and  the  plate.  The  repulsion  existing  between  the  liquid 
and  the  plate  is  usually  imputed  to  the  existence  of  an  atmos- 
phere of  vapour  upon  whichj  as  upon  a  cushion,  the  spheroids 
are  supposed  to  rest.  There  is  no  reason  to  conclude,  however, 
because  vapour  is  raised  from  a  liquid,  that  therefore  its  gravity 
must  be  suspended,  and  the  cause  is  rather  to  be  sought  for  in 
the  motion  of  the  spheroid,  or  of  its  internal  particles,  whereby 
the  motion  to  which  gravity  is  due  is  partially  counteracted. 


COMMUNICATION    OF   HEAT.  171 

Spheroidal  State  of  the  Water  in  Sailers.— There  can  be 
no  doubt  that  the  water  of  boilers  is  sometimes  repelled  from 
the  metal  in  the  same  manner  as  would  be  done  if  it  were  in 
the  spheroidal  state,  and  explosions  have,  no  doubt,  frequently 
had  their  origin  in  this  phenomenon.  Land  boilers,  whether  of 
the  cylindrical  or  waggon  form,  frequently  bend  down  in  the 
bottom  where  the  strongest  heat  of  the  furnace  impinges,  and 
lead  rivets,  inserted  in  them  for  purposes  of  safety,  are  some- 
times melted  out.  The  water  appears  to  be  repelled  from  the 
iron  in  those  parts  of  the  boiler  bottom  where  the  heat  is  great- 
est, and  the  iron  becomes  red  hot,  and  is  bagged  or  bent  out  by 
the  pressure  of  the  steam.  In  some  boilers  the  bottom  can  at 
any  time  be  made  red  hot  by  very  heavy  firing,  and  in  most  fac- 
tory boilers  the  bottom  will  be  more  or  less  injured  if  the  stoker 
urges  the  fire  very  much.  If  gauge  cocks  be  inserted  at  differ- 
ent levels,  in  a  small  upright  cylindrical  boiler,  so  that  one  cock 
is  near  the  top,  another  near  the  bottom,  and  the  rest  in  inter- 
mediate positions,  it  will  follow,  that  if  sufficient  water  be  intro- 
duced into  the  boiler  to  show  at  the  lowest  gauge  cock,  it  will 
continue  to  show  there  so  long  as  a  moderate  heat  is  maintained. 
So  soon,  however,  as  the  fire  is  made  to  burn  fiercely,  so  as  to 
impart  a  strong  heat  to  the  bottom,  the  water  will  disappear 
from  the  bottom  cock  and  show  in  the  top  cock,  thus  proving 
that  the  water  has  been  repelled  by  the  heat  until  it  occupies 
the  top  part  of  the  boiler  instead  of  the  bottom  part. 

COMMUNICATION  OF  HEAT. 

Heat  may  be  communicated  from  a  hot  body  to  a  cold  one  in 
three  ways — by  Eadiation,  by  Conduction,  and  by  Circulation. 

The  rapidity  with  which  heat  radiates  varies,  other  things 
being  equal,  as  the  square  of  the  temperature  of  the  hot  body  in 
excess  of  the  temperature  of  the  cold  one ;  so  that  a  body  if  made 
twice  as  hot  will  lose  a  degree  of  temperature  in  one-fourth  of 
the  time  ;  if  made  three  times  as  hot,  it  will  lose  a  degree  of 
temperature  in  one-ninth  of  the  time ;  and  so  on,  in  all  other 
proportions.  This  explains  how  it  comes  that  a  very  small  pro- 
portion of  surface  in  a  boiler  of  which  the  furnace  is  maintained 


172 


THEORY    OF   THE    STEAM-ENGINE. 


at  a  high  temperature  is  equivalent  to  a  much  larger  proportion 
of  surface  when  the  temperature  is  somewhat  lower.  Radiant 
heat  may  be  concentrated  into  a  focus  by  a  reflector,  in  the 
same  manner  as  light,  and,  like  light,  it  may  likewise  be  made 
to  undergo  retraction  and  polarisation. 

The  conduction  of  heat  through  different  substances  varies 
very  nearly  in  the  same  proportion  as  their  conducting  powers 
for  electricity.  Taking  the  conducting  power  of  silver  as  100, 
the  following  are  the  conducting  powers  of  metals  according  to 
the  best  authorities : — 

CONDUCTING   POWERS    OF    METALS. 


Name  of  Body. 

Conductivity  for  Electricity. 

Conductivity 
for  Heat. 

Eiess. 

Becquerel. 

Lenz. 

Wiedemann 
and  Franz. 

Silver  

100-0 
66-7 
69-0 
18-4 

10-0 

12-0 

100-0 

91-5 
64-9 

iV-6 

12-35 

100-0 
73-3 
58-5 
21-5 
22-6 
13-0 

100-0 

73-6 
63-2 
23-6 
14-5 
11-9 
11-6 
8-5 
8'4 
6-3 
1-8 

Copper  .  . 

Gold  

Brass  

Tin  

Iron  .  .  .  .  

Steel 

Lead  

7-0 
10-5 
5-9 

8-27 
7-93 

10-7 
10-3 

'  1-9 

Platinum  

German  Silver  

Bismuth  

The  conducting  power  of  marble  is  about  the  same  as  the 
conducting  power  of  bismuth ;  and  the  conducting  powers  of 
porcelain  and  bricks  are  each  about  half  that  of  marble.  The 
conducting  power  of  water  is  very  low,  and  hence  heat  is  trans- 
mitted downwards  through  water  only  very  slowly.  But  up- 
wards it  is  transmitted  rapidly  by  virtue  of  the  circulation  which 
then  takes  place. 

The  efficiency  of  the  heating  surface  of  a  boiler  will  depend 
very  much  upon  the  efficiency  of  the  arrangements  which  are 
in  force  for  maintaining  or  promoting  a  rapid  circulation  of  the 
water.  In  like  manner,  the  rapidity  of  the  circulation  which  is 
maintained  in  the  water  used  for  refrigeration  in  surface  con- 


CONDUCTING  POWERS  OF  DIFFERENT  SUBSTANCES.     173 

densers  will  mainly  determine  the  weight  of  steam  condensed 
in  the  hour  by  each  square  foot  of  refrigerating  surface.  Peclet 
found  by  a  number  of  experiments  that  water,  when  used  as  the 
refrigerating  fluid,  was  about  ten  times  more  effectual  than  air ; 
and  he  further  found  that  when  water  was  used  for  refrigera- 
tion, each  square  foot  of  copper  surface  was  able  to  condense 
about  21%  Ibs.  of  steam  in  the  hour.  Mr.  Joule,  however,  found 
that  a  square  foot  of  copper  surface  might,  by  maintaining  a 
rapid  circulation  of  the  cooling  water,  be  made  to  condense  100 
Ibs.  of  steam  in  the  hour — the  cooling  water  being  contained  in 
a  pipe  concentric  with  that  containing  the  steam,  and  flowing 
in  the  opposite  direction.  With  this  rapidity  of  refrigeration, 
the  cooling  surface  of  a  condenser  need  only  be  about  one  six- 
teenth of  the  heating  surface  of  the  boiler  which  supplies  the 
engine  with  steam.  In  ordinary  land  boilers  10  square  feet  of 
heating  surface  will  boil  off  a  cubic  foot,  or  62-J-  Ibs.  of  water  in 
the  hour ;  and  one  square  foot  of  heating  surface  will  therefore 
boil  off  one-tenth  of  this,  or  6'25  Ibs.  of  water  in  the  hour.  To 
boil  off  100  Ibs.  in  the  hour  would  at  this  rate  require  16  square 
feet  of  heating  surface.  But  the  100  Ibs.  of  steam  thus  boiled  off 
will,  according  to  Mr.  Joule,  be  condensed  by  one  square  foot  of 
cooling  surface ;  so  that,  if  this  authority  be  accepted,  the  surface 
of  a  well-constructed  condenser  need  only  be  about  one-sixteenth 
of  the  heating  surface  of  the  boiler,  the  steam  of  which  it  condenses. 
The  importance  of  maintaining  a  rapid  circulation  in  the 
water  of  boilers  has  not  yet  been  sufficiently  recognised.  It  is 
desirable  that  solid  water  and  not  steam  should  be  in  contact 
with  the  heating  surface,  else  the  metal  plating  will  be  liable  to 
become  overheated,  and  any  given  area  of  heating  surface  will 
be  much  less  effective.  The  species  of  boiler  invented  by  Mr. 
David  Napier,  called  the  haystack  boiler,  and  in  which  the 
water  is  contained  in  vertical  tubes,  is  about  the  best  species  of 
boiler  for  keeping  up  a  rapid  circulation  of  the  water.  But  it 
necessary  to  apply  large  return  pipes  or  a  wide  water  space  all 
round  the  exterior  of  the  boiler,  with  a  diaphragm  to  permit 
ascending  and  descending  currents,  in  order  that  the  water  car- 
ried upward  by  the  steam  may  be  immediately  returned. 


174  THEORY   OF   THE    STEAM-ENGINE. 

COMBUSTION. 

Combustion  is  energetic  chemical  combination  between  the 
oxygen  of  the  air  and  the  constituents  of  the  combustible.  The 
combustibles  chiefly  used  to  generate  the  heat  consumed  by 
steam-engines  are  coal,  wood,  and  sometimes  charcoal. 

Coal  consists  chiefly  of  carbon  and  hydrogen,  but  the  pro- 
portions in  which  these  elements  enter  into  the  composition  of 
different  coals  is  very  various.  Cannel  coal  consists  of  about  60 
per  cent,  of  volatile  matter,  and  40  per  cent,  of  coke  and  earthy 
matter,  whereas  splint  coal  consists  of  about  65  per  cent,  of 
coke,  and  35  per  cent,  of  volatile  matter.  Air  consists  of  oxy- 
gen and  nitrogen,  mixed  in  the  proportions  of  8  Ibs.  of  oxygen 
to  every  28  Ibs.  of  nitrogen,  or  1  Ib.  of  oxygen  to  every  3|  Ibs. 
of  nitrogen.  To  accomplish  the  combustion  of  6  Ibs.  of  carbon, 
16  Ibs.  of  oxygen  are  necessary,  forming  22  Ibs.  of  carbonic 
acid,  which  will  have  the  same  volume  as  the  oxygen,  and, 
therefore,  a  greater  density.  To  accomplish  the  combustion  of 
1  Ib.  of  hydrogen,  8  Ibs.  of  oxygen  are  necessary.  When,  there- 
fore, we  know  the  proportions  of  carbon  and  hydrogen  existing 
in  coal,  it  is  easy  to  tell  the  quantity  of  oxygen,  and,  conse- 
quently, the  quantity  of  air  necessary  for  its  combustion.  As  a 
general  rule,  it  may  be  stated  that,  for  every  pound  of  coal 
burned  in  a  furnace,  about  12  Ibs.  of  air  will  be  necessary  to 
furnish  the  oxygen  required,  even  if  every  particle  of  it  entered 
into  combination.  But  from  careful  experiments  it  has  been 
found,  that  in  ordinary  furnaces,  where  the  draught  is  produced 
by  a  chimney,  about  as  much  more  air  will  in  practice  be  neces- 
sary, or  about  24  Ibs.  per  Ib.  of  coal  burned.  In  the  case  of 
furnaces,  with  a  more  rapid  draught  maintained  either  by  a 
steam  jet  or  a  fan  blast,  a  smaller  excess  of  air  will  suffice,  and 
in  those  cases  about  18  Ibs.  of  air  will  be  required  from  the 
combustion  of  1  Ib.  of  coal.  If  a  cubic  foot  of  air  weigh  1-291 
oz.,  then  12  Ibs.  or  192  oz.  will  measure  about  150  cubic  feet,  as 
1-291  oz.  bears  the  same  proportion  to  1  cubic  foot,  as  192  oz. 
bears  to  150  cubic  feet  nearly.  In  ordinary  furnaces,  with  a 
chimney  therefor,  which  require  24  Ibs.  of  air  per  Ib.  of  coal, 


TOTAL    HEAT   PRODUCED    BY    COMBUSTION. 


175 


the  volume  of  air  necessary  for  the  combustion  of  1  Ib.  of  coal 
will  be  about  300  cubic  feet,  which  is  equal  to  the  content  of  a 
room  measuring  about  6  feet  8J  inches  every  way. 

The  specific  gravity  of  oxygen  is  a  little  more  than  that  of 
air,  being  by  the  latest  experiments  1-106,  while  that  of  air  is  1. 
Now,  as  16  Ibs.  of  oxygen  unite  with  6  Ibs.  of  carbon  to  form 
22  Ibs.  of  carbonic  acid,  and,  as  the  volume  of  the  carbonic  acid 
at  the  same  temperature  remains  only  the  same  as  that  of  the 
original  oxygen,  it  follows  that  the  density  or  specific  gravity  of 
the  carbonic  acid  must  be  greater  than  that  of  the  oxygen,  in 
the  same  proportion  in  which  22  is  greater  than  16.  Multiply- 
ing therefore  1-106,  which  is  the  specific  gravity  of  oxygen,  by 
22,  and  dividing  by  16,  we  get  1-521,  which  must  be  the  specific 
gravity  of  carbonic  acid,  if  the  specific  gravity  of  oxygen  is 
1-106.  Formerly,  the  specific  gravity  of  oxygen  was  reckoned 
at  1-111,  but  there  is  reason  to  believe  that  1-106  is  the  more 
accurate  determination. 

Total  Heat  of  Combustion. — The  temperature  to  which  a 
pound  of  fuel  would  raise  a  pound  of  water,  or  the  total  heat 
of  combustion  in  thermal  units,  has  been  carefully  investigated 
by  MM.  Favre  and  Silbennann,  whose  determinations  are  reca- 
pitulated and  condensed  by  M.  Rankine  as  follows : — 

TOTAL   HEAT   OF   COMBUSTION   OF   1    Ib.    OF   EACH   OF   THE 
COMBUSTIBLES   ENTJMEBATED. 


Combtntlble, 
I  Ib.  of  each  being  burned. 

Lbe.  of 
Air 
required. 

Lb«.  of 
Air 
required. 

Total  Heat 

in  Thermal 
Unit*. 

Evaporative 
Power 
from  212°. 

Hydrogen  «cas  

8 

86 

62082 

64-2 

Carbon,  imperfectly  burned,  ) 
so  as  to   make   carbonic  > 
oxide  j 

n 

6 

4,400 

4-55 

Carbon,  completely  burned,  I 
BO  as  to    make  carbonic  v 
acid  \ 

2! 

12 

14,500 

15-0 

Olettant  gas  

3J 

15J 

21844 

22-1 

Various  liquid  hydrocarbons  .  . 
Carbonic  oxide,  as  much  as") 
Is  made  by  the  imperfect  1 
combustion  of  1  Ib.  of  car-  f 
bon,  viz.  24  Ibs  J 

li 

6 

j  from  21,000 
(       to  19,000 

10,100 

(from  22 
\      to20 

10-45 

176  THEORY   OF   THE    STEAM-ENGINE. 

"With  regard  to  the  quantities  stated  as  being  the  total  heat 
of  combustion  respectively  of  carbon  completely  burned,  carbon 
imperfectly  burned,  and  carbonic  oxide,  Mr.  Eankine  says  that 
the  following  explanation  has  to  be  made : — 

The  burning  of  carbon  is  always  complete  at  first ;  that  is  to 
say,  one  pound  of  carbon  combines  with  2$  Ibs.  of  oxygen,  and 
makes  3f  Ibs.  of  carbonic  acid ;  and  although  the  carbon  is  solid 
immediately  before  the  combustion,  it  passes  during  the  com- 
bustion into  the  gaseous  state,  and  the  carbonic  acid  is  gaseous. 
This  terminates  the  process  when  the  layer  of  carbon  is  not  so 
thick,  and  the  supply  of  air  not  so  small,  but  that  oxygen  in 
sufficient  quantity  can  get  direct  access  to  all  the  solid  carbon. 
The  quantity  of  heat  produced  is  14,500  thermal  units  per  Ib. 
of  carbon,  as  already  stated.  . 

But  in  other  cases  part  of  the  solid  carbon  is  not  supplied 
directly  with  oxygen,- but  is  first  heated,  and  then  dissolved  into 
the  gaseous  state,  by  the  hot  carbonic  acid  gas  from  the  other 
parts  of  the  furnace.  The.  3f  Ibs.  of. carbonic,  acid  gas  from  1 
Ib.  of  carbon,  are  capable  of  dissolving  an  additional  Ib.  of  car- 
bon, making  4f  Ibs.  of  carbonic .  oxide  -gas ;  and  the  volume  of 
this  gas  is  double  of  that  of  the  carbonic  acid  gas  which  pro- 
duces it.  In  this  case,  the  heat  produced,  instead  of  being  that 
due  to  the  complete  combustion  of 

1  Ib.  of  carbon  or     ...  .  1  ,         •  .        14,500 

falls  to  the  amount  due  to  the  imperfect  combustion  of  2  Ibs. 

of  carbon,  or  ..."  '." .     2x4,400  x  8,800 

Showing  a  loss  of  heat  to  the  amount  of        .  V'"    ""'.'    J      5,700 

which  disappears  in  volatilising  the  second  pound  of  carbon. 
Should  the  process  stop  here,  as  it  does  in  furnaces  ill  supplied 
with  air,  the  waste  of  fuel  is  very  great,  as  the  carbonic  oxide — 
which  is  a  species  of  invisible  smoke — has  a  large  quantity  of 
carbon  in  it  which  is  dissipated  in  the  atmosphere  without  use- 
ful result.  But  when  the  4$  Ibs.  of  carbonic  oxide  gas,  contain- 
ing 2  Ibs.  of  carbon,  is  mixed  with  a  sufficient  supply  of  fresh 
air,  it  burns  with  a  blue  flame,  combining  with  an  additional  2f 
Ibs.  of  oxygen,  making  7£  Ibs.  of  carbonic  acid  gas,  and  giving 


ECONOMIC    VALUES    OF   DIFFERENT   COALS. 


177 


additional  heat  of  double  the  amount  due  to  the  combustion  of 
1-J-  Ib.  of  carbonic  oxide  ;  that  is  to  say, 

10,100  x  2  =  20,200 
to  which  being  added  the  heat  produced  by  the  imperfect 

combustion  of  2  Ibs.  of  carbon,  or  ...         8,800 

there  is  obtained  the  heat  due  to  the  complete  combustion  of 

2  Ibs.  of  carbon,  or      .  .  .  .  2  x  14,500  =  29,000 

The  evaporative  powers  of  different  kinds  of  coal  in  practice 
is  given  in  the  following  table  : — 

TABLE   SHOWING  THE  ECONOMIC   VALUES   OF   DIFFEBENT   COALS. 

BY  DE  J.A.  BECIIE  AND  PLAYFAIK. 


I 

fames  of  Coal  employed  In  the 
Experiment*. 

Economical 
evaporating 

her  of  Ibs.,  of 
Water  evapo- 
rated from 
21  2«  by  1  Ib.  of 
CoaL 

Weight  of 
1  cubic  foot 
of  the  Coal 
as  used  for 
Fuel. 

Ibs. 

Space  occn- 
pied  by  1  ton 
of  the  Coal 
la  cubic  feet 

Rate  of  eva- 
poration, or 
number  of 
Ibs.  of  Water 
evaporated 
per  hour. 

Mean. 

f  Graigola  

9-35 

60-166 

37-28 

441-48 

Anthracite  (Jones  &  Co.) 
Oldcastle  Fiery  Vein  
Ward's  Fiery  Vein  
Binea  

9-46 
8-94 
9-40 
9-94 

58-25 
50-916 
57-483 
57-08 

88-45 
43-99 
89 
89-24 

409-37 
464-30 
629-90 
486-95 

Llangennech  

8-86 

56-93 

89-34 

873-22 

Pentrepoth  

8'72 

57-72 

38-80 

881-50 

Pentrelellin  

6-36 

66-166 

83-85 

247-24 

1 

Duffryn  

10-14 

58-22 

42-09 

409-32 

— 

Mynydd  Newydd  

9-52 

56-83 

89-76 

470-69 

•= 

Threft-quarter'Rock  Vein 
Cwm  Frood  Eock  Vein. 
Cwm  Nanty-gros  

8-84 
8-70 

8-42 

66-388 
55-277 
56-0 

89-72 
40-52 
40-00 

486-86 
879-80 
404-16 

Kesolven  

9-53 

58-66 

88-19 

890-25 

7-47 

55-7 

40-216 

250-40 

Bed  was  

9-79 

60-5 

44-32 

476-96 

Ebbw  Vale  

10-21 

58-3 

42-26 

460-22 

Porthmawr  

7-53 

58-0 

42-02 

847-44 

LColeshill  

8-00 

53-0 

42-26 

406-41 

^ 
s 

- 

'Dalkeith  Jewel  Seam... 
"       Coronation      ( 
Seam  ) 
Wallsend  Elgin  

7-08 
7-71 
8-46 

498 
51-66 

M-ii 

44-98 
48-86 
41-02 

855-18 
870-08 
435-77 

rli 

Fordel  Splint  

7-56 

55-0 

40-72 

464-98 

Grangemouth  

7-40 

54-25 

40-18 

880-49 

$ 

1  '.mi  mill  ill   

7-80 

52'5 

42-67 

897-78 

=    - 

H 

Lydney  (Forest  of  Dean) 

Slievardagh  (Irish  An-  I 
thracite)  j 

8-52 
9-85 

54-444 
62-8 

41-14 
85-66 

487-19 
473-18 

"Wylam's  Patent  Fuel... 
Warlich'8            " 
Bell's 

8-92 
10-86 
8-58 

65-08 
69-05 
65-8 

84-41 
82-44 
84-80 

418-89 
457-84 
649-11 

8* 


178  THEORY    OF   THE    STEAM-ENGINE. 

Maximum  Temperature  of  the  Furnace. — When  we  know 
the  total  heat  of  a  combustible  in  thermal  units,  the  weight  of 
the  smoke  and  ashes  or  the  products  of  combustion,  as  they  are 
called,  and  their  specific  heat,  it  is  easy  to  tell  what  is  the  high- 
est temperature  that  the  furnace  can  attain,  supposing  that  the 
air  is  not  artificially  heated.  Thus  the  chief  products  of  combus- 
tion of  coal  being  carbonic  acid,  steam,  nitrogen,  and  ashes,  with 
a  certain  proportion  of  residual  air,  which  passes  unchanged 
through  the  fire ;  then,  if  we  reckon  the  specific  heat  of  carbonic 
acid  at  0-217,  of  steam  at  0*475,  of  nitrogen  at  0-245,  of  air  at 
0*238,  and  of  ashes  at  0*200,  and  take  into  account  the  quantities 
of  each  which  are  present,  the  mean  specific  heat  of  the  prod- 
ucts of  combustion  may  be  taken,  without  much  error,  as  about 
equal  to  the  specific  heat  of  air.  Now,  as  12  Ibs.  of  air  are  re- 
quired for  the  combustion  of  a  pound  of  carbon,  even  if  every 
particle  of  the  oxygen  be  supposed  to  enter  into  combination, 
the  weight  of  the  products  of  combustion  will  on  that  supposi- 
tion be  12  Ibs.  +  1  lb.,  or  13  Ibs.  If  we  take  the  total  heat  of 
combustion  of  carbon  or  charcoal  at  14,500,  and  the  mean  speci- 
fic heat  of  the  products  of  combustion  at  0-238,  then  the  specific 
heat  multiplied  by  the  weight  will  be  3*094 ;  and  14,500  divided- 
by  3*094  =  4689,  which  will  be  the  temperature  to  which  the 
furnace  would  be  raised  in  degrees  Fahrenheit,  supposing  every 
atom  of  oxygen  that  entered  the  furnace  entered  into  com- 
bination. If,  however,  as  will  be  the  case  in  ordinary  furnaces, 
twice  that  quantity  of  air  necessarily  enters,  then  the  weight  of 
the  products  of  combustion  of  1  lb.  of  coal  will  be  25  lb.,  which, 
multiplied  by  the  specific  heat  =  5*95,  and  14,500  divided  by 
5-95  =  2,437,  which  is  the  temperature  in  degrees  Fahrenheit 
that,  on  this  supposition,  the  furnace  would  have.  If  18  Ibs.  of 
air  be  supplied  per  lb.  of  coal,  as  suffices  in  the  case  of  furnaces 
with  artificial  draught,  then  the  weight  of  the  products  of  com- 
bustion will  be  19  Ibs.,  which,  multiplied  by  the  specific  heat, 
gives  4-522,  and  14,500  divided  by  4-522,  gives  3,207  as  the  tem- 
perature of  the  furnace  in  degrees  Fahr.  This  in  point  of  fact 
may  be  taken  as  a  near  approach  to  the  temperature  of  hot  fur- 
naces, such  as  that  of  a  locomotive  boiler. 


RATE   OP   COMBUSTION. 


179 


The  increased  volume  which  any  given  quantity  of  air  at  32° 
will  acquire,  by  raising  its  temperature  through  any  given  num- 
ber of  degrees,  can  easily  be  determined  by  the  rule  already 
given  for  that  purpose.  Mi*.  Rankine  has  computed  the  volume 
in  cubic  feet,  which  121bs.  of  air,  18  Ibs.,  and  24  Ibs.,  will  respec- 
tively acquire,  when  heated  to  different  temperatures,  by  com- 
bining with  1  Ib.  of  carbon  in  a  furnace ;  the  volume  of  12  Ibs.  at 
32°,  and  at  the  atmospheric  pressure,  being  taken  at  150  cubic 
feet,  of  18  Ibs.  at  225  cubic  feet,  and  of  24  Ibs.  at  300  cubic  feet. 
The  results  are  as  follows : 


TEMPERATURES   OF   COMBUSTION  AND   VOLUMES   OP    PRODUCTS. 


Supply  of  Air  In  pounds  per  Ib.  of  fuel. 

Temperatures. 

12  Ibs. 

18  Ibs. 

24  Ibs. 

Volume  of  Air  or  Gases  In  cubic  feet  at  each 

Temperature. 

32° 

150 

225 

300 

68° 

161 

241 

322 

104° 

172 

258 

344 

212° 

205 

307 

409 

892° 

259 

389 

519 

572° 

314 

471 

628 

752° 

369 

553 

738 

1112° 

479 

718 

957 

1472° 

588 

882 

1176 

1832° 

697 

1046 

1395 

2500° 

906 

1359 

1812 

3275° 

1136 

1704 

4640° 

1551 

Rate  of  Combustion. — The  rate  of  combustion,  or  the  quan- 
tity of  fuel  burned  in  the  hour  upon  each  square  foot  of  fire- 
grate, varies  very  much  in  different  classes  of  boilers.  In  Cor- 
nish boilers  it  is  3£  Ibs.  per  square  foot ;  in  the  older  class  of 
land  boilers,  10  Ibs. ;  in  more  recent  land  boilers,  13  to  14  Ibs. ; 
in  modern  marine  boilers,  16  to  24  Ibs.,  and  in  locomotive  boilers* 
80  to  120  Ibs.  on  each  square  foot  of  fire-grate  in  the  hour. 


180  THEORY   OF   THE   STEAM-ENGINE. 


THERMO-DYNAMICS. 

It  has  been  already  stated  that  heat  and  power  are  mutually 
convertible,  and  that  the  power  in  the  shape  of  heat  which  is 
necessary  to  raise  a  pound  of  water  through  one  degree  Fahren- 
heit, would,  if  utilised  without  waste  in  a  thermo-dynamic  en- 
gine, raise  7T2  Ibs.  through  the  height  of  1  foot.  A  pound  of 
water  raised  through  a  degree  centigrade  is  equivalent  to  1390  Ibs. 
raised  through  the  height  of  1  foot.  In  every  heat  engine,  the 
greater  the  extremes  of  temperature,  or  the  hotter  the  boiler  or 
source  of  heat  and  the  colder  the  condenser  or  refrigerator,  the 
larger  will  be  the  proportion  of  the  heat  utilised  as  power. 

In  a  perfect  steam  engine,  if  a  be  the  temperature  of  the 
boiler,  reckoning  from  the  point  of  absolute  zero,  and  &  be  the 
temperature  of  the  condenser,  reckoning  also  from  the  point  of 
absolute  zero,  the  fraction  of  the  entire  heat  communicated  to 
the  boiler  which  will  be  converted  into  mechanical  effect,  will 

be  —  — .   Now  it  is  clear  if  a  =  5,  or  if  the  temperature  of  the 

a  a j 

boiler  and  condenser  are  the  same,  the  value  of becomes 

a 

equal  to  0,  or  there  is  none  of  the  heat  utilised  as  power,  whereas, 
on  the  other  hand,  if  a  be  taken  larger  and  larger,  the  value  of 
the  fraction  becomes  continually  greater,  indicating  that  by  in- 
creasing the  difference  of  the  temperatures  of  the  boiler  and 
condenser,  a  great  quantity  of  the  heat  expended  is  converted 
into  mechanical  effect,  and  by  taking  a=  o>,  or  infinity,  the  limit 
to  which  the  fraction  approaches  is  found  to  be  unity,  showing 
that  in  such  a  case,  if  it  were  possible  of  realisation,  the  whole 
of  the  heat  would  be  converted  into  power. 

The  formula  given  by  Professor  Thomson  for  determining 
the  power  generated  by  a  perfect  thermo-dynamic  engine,  is  as 
follows : — 

If  S  be  the  temperature  of  the  source  of  heat,  and  T  be  the 
temperature  of  the  refrigerator,  both  expressed  in  centigrade 
degrees ;  and  if  H  denote  the  total  heat  in  thermal  units  centi- 
grade, entering  the  engine  in  a  given  time ;  and  J  be  Joule's 


POWER   PRODUCIBLE    IN    A   PERFECT    ENGINE.          181 

equivalent  of  1390  Ibs.  raised  one  foot  high  by  a  centigrade  de- 
gree ; — then  the  power  produced,  or  "W  the  work  performed,  is 

S— T 
W=JH-    -' 

S  +  274 

This  formula  may  be  expressed  in  words,  as  follows : — 

TO  FIND  THE  POWEE  GENERATED  BY  A  PERFECT  ENGINE  IMPELLED 
BY  THE  MOTIVE  POWER  OF  HEAT. 

KFLE. — From  the  temperature  of  the  source  or  ~boiler,  subtract 
the  temperature  of  the  condenser ;  divide  the  remainder  ty 
the  sum  of  the  temperature  of  the  source  and  274,  and  multi- 
ply the  quotient  ty  the  total  heat  communicated  to  the  en- 
gine per  minute,  expressed  in  the  number  of  degrees  through 
which  it  would  raise  one  pound  of  water.  Finally,  multiply 
this  product  by  1390.  The  result  is  the  number  of  pounds 
that  the  engine  will  raise  a  foot  high  in  the  minute.  The 
temperatures  are  all  taken  in  degrees  centigrade. 

Example. — In  a  steam-engine  working  with  a  pressure  of  14 
atmospheres,  the  temperature  of  the  steam  in  the  boiler  will  be 
215°  centigrade,  and  the  temperature  of  the  condenser  may  be 
taken  at  44-44°  centigrade.  If  a  gram  of  coal  be  burned  per 
minute,  the  heat  imparted  every  minute  to  a  pound  of  water 
win  be  -905°  centigrade.  Now  215  —  44-44°  =  170-56  and 
215  +  274  =  489,  and  170-56  divided  by  489  =  0-35,  which  mul- 
tiplied by  -905  and  by  1390  =  440  Ibs.  raised  1  foot  high  every 
minute,  which  as  a  grain  of  coal  is  burned  every  minute,  is  very 
nearly  the  same  result  as  that  before  indicated. 

Cheapest  Source  of  Motive  Power. — The  cheapest  source  of  a 
mechanical  power  that  will  be  available  in  all  situations,  is,  so 
far  as  we  yet  know,  the  combustion  of  coal.  Electricity  and 
galvanism  have  been  proposed  as  motive  powers,  and  may  be 
used  as  such,  but  they  are  much  more  expensive  than  coal.  Mr. 
Joule  has  ascertained  by  his  experiments  that  a  gram  of  zinc, 
consumed  in  a  galvanic  battery,  will  generate  sufficient  power  to 
raise  a  weight  of  145-6  Ibs.  through  the  height  of  one  foot; 


182  THEORY   OF   THE    STEAM-ENGINE. 

whereas  a  grain  of  coal,  consumed  by  combustion,  will  generate 
sufficient  power  to  raise  1261-45  Ibs.  to  the  height  of  1  foot. 

Moreover,  it  appears  certain  that  Mr.  Joule's  estimate  of  the 
heating  power  of  coal  is  too  small.  A  pound  of  coal  will,  under 
favourable  circumstances,  evaporate  12  Ibs.  of  water,  which  is 
equivalent  to  a  pound  of  water  being  heated  2  degrees  Fahren- 
heit by  a  grain  of  coal,  or  it  is  equivalent  to  1544  Ibs.  raised 
through  1  foot.  This  is  more  than  ten  times  the  power  gener- 
ated by  a  pound  of  zinc.  But  as  thermo-electric  engines,  it  is 
estimated,  expend  their  energy  about  four  times  more  bene- 
ficially than  heat  engines,  the  dynamic  efficacy  of  a  pound  of 
zinc  may  be  taken  as  about  4-10ths  of  that  of  a  pound  of  coal.  A 
ton  of  zinc,  however,  costs  fifty  or  sixty  times  as  much  as  a  ton 
of  coals,  while  it  is  not  half  so  effective.  There  does  not  ap- 
pear, therefore,  to  be  the  least  chance  of  heat  engines  being 
superseded  by  electro-dynamic  engines,  of  which  zinc  or  some 
other  metal  supplies  the  motive  force. 

EXPANSION    OF    STEAM. 

When  air  is  compressed  into  a  smaller  volume,  a  certain 
amount  of  power  is  expended  in  accomplishing  the  compression, 
which  power,  as  in  the  case  of  a  bent  spring,  is  given  back  again 
when  the  pressure  is  withdrawn.  If,  however,  the  air  when 
compressed  is  suddenly  dismissed  into  the  atmosphere,  the  power 
expended  in  compression  will  be  lost ;  and  there  is  a  loss  of 
power,  therefore,  in  dispensing  with  that  power,  which  is  re- 
coverable by  the  expansion  of  the  air  to  ita  original  volume. 
Now  the  steam  of  an  engine  is  in  the  condition  of  air  already 
compressed;  and  unless  the  steam  be  worked  in  the  cylinder 
expansively — which  is  done  by  stopping  the  supply  from  the 
boiler  before  the  stroke  is  closed — there  will  be  a  loss  of  a  cer- 
tain proportion  of  the  power  which  the  steam  would  otherwise 
produce.  If  the  flow  of  steam  to  an  engine  be  stopped  when  the 
piston  has  performed  one-half  of  the  stroke,  leaving  the  rest  of 
the  stroke  to  be  completed  by  the  expanding  steam,  then  the 
efficacy  of  the  steam  will  be  increased  1-7  times  beyond  what  it 


MODE    OF   COMPUTING   BENEFIT   OF   EXPANSION.       183 

would  have  been  had  the  steam  at  half-stroke  been  dismissed 
•without  extracting  more  power  from  it ;  if  the  steam  be  stopped 
at  one-third  of  the  stroke,  the  efficacy  will  be  increased  2-1 
times;  at  one-fourth,  2'4  times;  at  one-lifth,  2'6  times;  at  one- 
sixth,  2 -8  times;  at  one- seventh,  3  times;  and  at  one-eighth,  3*2 
times. 

TO   FIND   THE    INCREASE    OF    EFFICIENCY   ARISING   FROM   WORKING 
STEAM   EXPANSIVELY. 

RULE. — Divide  the  total  length  of  the  stroke  ~by  the  distance 
(which  call  1)  through  which  the  piston  moves  before  the  steam 
is  cut  off.  The  Neperian  logarithm  of  the  whole  stroke  ex- 
pressed in  terms  of  the  part  of  the  strolce  performed  with  the 
full  pressure  of  steam,  represents  the  increase  of  efficiency 
due  to  expansion. 

Example  1. — Suppose  that  the  steam  be  cut  off  at  ^  ths  of 
the  stroke :  what  is  the  increase  of  efficiency  due  to  expansion  ? 

Here  it  is  plain  that  -j-^ths  of  the  whole  stroke  is  the  same  as 
T?T  of  the  whole  stroke.  The  hyperbolic  logarithm  of  7'5  is 
2*015,  which  increased  by  1,  the  value  of  the  portion  performed 
with  full  pressure,  gives  3'015  as  the  relative  efficacy  of  the 
steam  when  expanded  to  this  extent,  instead  of  1,  which  would 
have  been  the  efficacy  if  there  had  been  no  expansion. 

If  the  steam  be  cut  of  at  |,  f ,  f ,  |,  f ,  £ ,  or  $th  of  the  stroke, 
the  respective  ratios  of  expansion  will  be  8,  4,  2-66,  2,  1-6,  1'33, 
and  1-14,  of  which  the  respective  hyperbolic  logarithms  are 
2-079,  1-386,  0-978,  0-693,  0*470,  0-285,  and  0-131 ;  and  if  the 
steam  be  cut  off  at  TV,  -?*,  T3«,  T\>  tV»  TIT>  T7i»  A,  or  rVhs  of  the 
stroke,  the  respective  ratios  of  expansion  will  be  10,  5,  3-33,  2-5, 
1-66,  1'42,  1-25,  and  1-11,  of  which  numbers  the  respective 
hyperbolic  logarithms  are  2-303, 1-609,  1-203,  0-916,  0-507,  0-351, 
0'223,  and  0'104.  With  these  data  it  will  be  easy  to  compute 
the  mean  pressure  of  steam  of  any  given  initial  pressure  when 
cut  off  at  any  eighth  part  or  any  tenth  part  of  the  stroke,  as  we 
have  only  to  divide  the  initial  pressure  of  the  steam  in  Ibs.  per 
square  inch  by  the  ratio  of  expansion,  and  to  multiply  the  quo- 


184  THEORY    OF   THE    STEAM-ENGINE. 

tient  by  the  hyperbolic  logarithm,  increased  by  1,  of  the  number 
representing  the  ratio,  which  gives  the  mean  pressure  through- 
out the  stroke  in  Ibs.  per  square  inch.  Thus,  if  steam  of  100  Ibs. 
be  cut  off  at  half  stroke,  the  ratio  of  expansion  is  2,  and  100 
divided  by  2  and  multiplied  by  T693  =  84'65,  which  is  the  mean 
pressure  throughout  the  stroke  in  Ibs.  per  square  inch.  The 
terminal  pressure  is  found  by  dividing  the  initial  pressure  by  the 
ratio  of  expansion;  thus,  the  terminal  pressure  of  steam  of 
100  Ibs.  cut  off  at  half  stroke  will  be  100  divided  by  2  =  50  Ibs. 
per  square  inch. 

Example  2. — "What  is  the  mean  pressure  throughout  the 
stroke  of  steam  of  200  Ibs.  per  square  inch  cut  off  at  ~th  of  the 
stroke  ? 

Here  200  divided  by  10  =  20,  which,  multiplied  by  3-303  (the 
hyperbolic  logarithm  of  10  increased  by  1)  gives  66'04,  which  is 
the  mean  pressure  exerted  on  the  piston  throughout  the  stroks 
in  Ibs.  per  square  inch. 

If  the  steam  were  cut  off  at  ]th  of  the  stroke  instead  of  ^Vth, 
then  we  should  have  200  divided  by  8  =  25,  which,  multiplied 
by  3'079  (the  hyperbolic  logarithm  of  8  increased  by  1),  gives 
TS'O'TS  Ibs.,  which  would  be  the  mean  pressure  on  the  piston 
throughout  the  stroke  in  such  a  case. 

If  the  initial  pressure  of  the  steam  were  3  Ibs.  per  square 
inch,  and  the  expansion  took  place  throughout  £ths  of  the  stroke, 
or  the  steam  were  cut  off  at  |th,  then  3  -j-  8  =  '375,  which 
x  by  3-079  =  1'154625  Ibs.  per  square  inch  of  mean  pressure. 

There  are  various  expedients  for  stopping  off  the  supply  of 
steam  to  the  engine  at  any  desired  point  of  the  stroke,  which 
are  described  in  my  '  Catechism  of  the  Steam  Engine,'  and 
which,  consequently,  it  would  be  superfluous  to  recapitulate 
here.  One  mode  is  by  the  use  of  an  expansion  valve,  and 
another  mode  is  by  extending  the  length  of  the  face  of  the  or- 
dinary slide  valve  by  which  the  steam  is  let  into  and  out  of  the 
cylinder,  which  extension  of  the  face  is  called  lap  or  cover. 
For  the  purposes  of  this  work  it  will  be  sufficient  to  recapitu- 
late the  mean  pressure  of  the  steam  on  the  piston  of  an  engine 
throughout  the  whole  stroke,  supposing  the  steam  to  be  cut  off 


PRESSURES  AT  DIFFERENT  RATES  OF  EXPANSION.       185 

at  different  successive  points  of  the  stroke,  counting  first  by 
eighths,  and  next  by  tenths,  and  to  explain  what  amount  of  lap 
answers  to  a  given  expansion,  and  what  expansion  follows  the 
use  of  a  given  proportion  of  lap.  The  mean  pressure  of  the 
steam  throughout  the  stroke,  with  different  initial  pressures  of 
steam  and  different  rates  of  expansion,  or,  in  other  words,  the 
equivalent  constant  pressure  that  would  be  exerted  throughout 
the  stroke  if  such  a  pressure  were  substituted  for  the  varying 
pressures  to  which  the  piston  is  in  reality  subjected,  are  exhib- 
ited in  the  following  tables,  in  one  of  which  the  pressures  are 
those  which  would  ensue  if  the  expansion  took  place  during  so 
many  eighths  of  the  stroke,  and  in  the  other  during  so  many 
tenths  of  the  stroke : — 


MEAN  PRESSURE   OF   STEA.M  AT  DIFFERENT  DENSITIES  AND  BATES 
OF  EXPANSION. 

The  column  headed  0,  contains  the  Initial  Pressure  in  Ibs.,  and  the 
remaining  columns  contain  the  Mean  Pressure  in  Ibs.,  with  different 
amounts  of  Expansion. 


Proportion  of  the  Stroke  through  which  Expansion  takes  place. 

0 

i 

» 

s 

1 

£ 

I 

V 

8 

2-96 

2-89 

2-75 

2-53 

2-22 

1-789 

1-154 

4 

8-95 

8-85 

8-67 

8-38 

2-96 

2-386 

1-589 

5 

4-948 

4-S18 

4-598 

4-232 

8-708 

2-982 

1-921 

6 

6-987 

5-782 

5-512 

5-079 

4-450 

8-579 

2-309 

7 

6-927 

6-746 

6-431 

5-925 

5-241 

4175 

2-694 

8 

7-917 

7-710 

7-350 

6-772 

6-984 

4-772 

8-079 

9 

8-906 

8-678 

8-268 

7-618 

6-675 

6-868 

8-463 

10 

9-896 

9-637 

9-187 

8-465 

7-417 

5-965 

3-848 

11 

10-885 

10-601 

10-106 

9-311 

8-159 

6-561 

4-288 

12 

11-875 

11-565 

10-925 

10-158 

8-901 

7-158 

4-618 

18 

12-865 

12-528 

11-948 

11-004 

9-642 

7-754 

6-008 

14 

18-854 

13-492 

12-862 

11-851 

10-884 

8-581 

r>-:!ss 

15 

14-844 

14-456 

13-781 

12-697 

11-126 

8-947 

5-778 

16 

15-834 

15-420 

14-700 

13-544 

11-868 

9-544 

6-158 

17 

16-828 

16-883 

15-618 

14-890 

12-609 

10-140 

6-542 

18 

17-818 

17-847 

16-587 

16-287 

18-851 

10-787 

6-927 

19 

18-702 

18-811 

17-448 

16803 

14098 

11-338 

7-812 

20 

19-792 

19-275 

18-875 

17-970 

14-885 

11-930 

7-697 

25 

24-740 

24-093 

22-968 

21-162 

18-548 

14-912 

9-621 

80 

29-688 

28-912 

27'5«2 

25-895 

22-252 

17-895 

11-546 

85 

84-686 

88-781 

88-156 

29-627 

25-961 

20-877 

18-470 

40 

39-585 

8S-550 

86-750 

88-860 

29-670 

23-860 

15-895 

45 

44-588 

48-368 

41-348 

88-092 

88-878 

26-842 

17-819 

50 

49-481 

48-187 

45-987 

42-825 

87-067 

29-825 

19-248 

186 


THEOKY   OF   THE    STEAM-ENGINE. 


MEAN  PEESSURE   OF   STEAM   AT   DIFFERENT    DENSITIES  AND  EATE3 
OF   EXPANSION. 

The  column  Jieaded  0  contains  the  Initial  Pressure  in  Ibs.,  and  the 
remaining  columns  contain  the  Mean  Pressure  in  Ibs.,  with  different 
amounts  of  Expansion. 


Proportion  of  the  Stroke  through  which  Expansion  takes  place. 

Q 

4 

5 

g 

1  O 

1  U 

1  0 

1  0 

10 

10 

1  0 

To 

10 

'A 

2-980 

2-930 

2-830 

2-710 

2-539 

2-299 

1-981 

1-668 

0-990 

4 

3-974 

3-913 

3-780 

3-614 

3-3S6 

3-065 

2-642 

2-087 

1-320 

5 

4-968 

4-892 

4-725 

4-518 

4-232 

8-832 

8-303 

2-609 

1-651 

6 

5-961 

5-870 

5-670 

5-421 

5-079 

4-598 

3-963 

8-130 

1-981 

7 

6-955 

6-848 

6-615 

6-325 

5-925 

5-364 

4-624 

8-652 

2-811 

8 

7-948 

7-827 

7-560 

7-228 

6-772 

6-131 

5-284 

4-174 

6-641 

9 

8-942 

8-805 

8-505 

8-132 

7-618 

6-897 

5-945 

4-696 

2-971 

10 

9-936 

9-784 

9-450 

9-036 

8-465 

7-664 

6-606 

5-218 

8-302 

11 

10-929 

10-762 

10-395 

9-939 

9-311 

8-430 

7-266 

5-739 

3-632 

12 

11-923 

11-740 

11-340 

10-843 

10-158 

9-196 

7-927 

6-261 

8-962 

13 

12-856 

12-719 

12-285 

11-746 

10-994 

9-963 

8-587 

6-783 

4-292 

14 

IS  310 

13-967 

13-230 

12-650 

11-851 

10-729 

9-248 

7-305 

4-622 

15 

14-904 

14-676 

14-175 

13-554 

12-697 

11-496 

9-909 

7-827 

4-953 

16 

15-897 

15-654 

15-120 

14-457 

13-544 

12-262 

10-569 

8-348 

5-283 

17 

16-891 

16-632 

16-065 

15-361 

14-051 

13-028 

11-230 

8-870 

5-613 

18 

17-884 

17-611 

17-010 

16-264 

15-237 

13-795 

11-890 

9-392 

5-944 

19 

18-878 

18-589 

17-955 

17-168 

16-083 

14-561 

12-551 

9-914 

6-273 

20 

19-872 

19-568 

18-900 

18-072 

16-930 

15-328 

18-212 

10-436 

6-600 

25 

24-840 

24-460 

23-625 

22-590 

21-162 

19-100 

16-515 

13-040 

8-255 

30 

29-808 

29-352 

28-350 

27-108 

25-895 

22-992 

19-818 

15-654 

9-006 

85 

84-776 

34-244 

88-075 

81-626 

29-627 

26-824 

23-121 

18-263 

11-557 

40 

39-744 

39-136 

37-800 

36-144 

33-860 

30-656 

26-224 

20-872 

13-208 

45 

44-912 

44-028 

42-525 

40-662 

88-0921  34-888 

29-727 

23-481 

14-859 

50 

49-630 

48-920 

47-250 

45-180 

42-325:  38-320 

83-030 

26-090 

16-510 

Example. — If  steam  be  admitted  to  the  cylinder  at  a  pressure 
of  3  Ibs.  per  square  inch,  and  be  suffered  to  expand  during  -Jth  of 
the  stroke,  the  mean  pressure  during  the  whole  stroke  will  be 
2*96  Ibs.  per  square  inch.  In  like  manner,  if  steam  at  the  press- 
ure of  3  Ibs.  per  square  inch  were  cut  off  after  the  piston  had 
gone  through  the  ^th  of  the  stroke,  leaving  the  steam  to  expand 
through  the  remaining  £ths,  the  mean  pressure  during  the  whole 
stroke  would  be  1*154  Ibs.  per  square  inch. 


EELATIONS  BETWEEN  THE    LAP  OF  THE  VALVE  AND  THE  AMOUNT 
OF  EXPANSION. 

The  rules  for  determining  the  relations  between  the  lap  of 
the  valve  and  the  amount  of  the  expansion  are  as  follows : — 


EFFECTS    OF   LAP   ON   THE    VALVE.  187 

TO  FIND  HOW  MUCH  LAP  MUST  BE  GIVEN  ON  THE  STEAM  SIDE, 
IN  ORDER  TO  CUT  THE  STEAM  OFF  AT  ANY  GIVEN  PAET  OF 
THE  6TBOKE. 

RULE. — From  the  length  of  the  strode  of  the  piston  subtract  the 
length  of  that  part  of  the  stroke  that  is  to  be  made  before 
the  steam  is  cut  off.  Divide  the  remainder  by  the  length  of 
the  stroke  of  the  piston,  and  extract  the  square  root  of  the 
quotient.  Multiply  the  square  root  thiis  found  by  half  the 
length  of  the  stroke  of  the  valve,  and  from  the  product  take 
half  the  lead,  and  the  remainder  will  be  the  amount  of  lap 
required. 

TO   FIND   AT   WHAT   PAET   OF   THE   STEOKE   ANY  GIVEN   AMOUNT  OF 
LAP   ON   THE   STEAM   SIDE   WILL   CUT   OFF   THE   STEAM. 

RULE. — Add  the  lap  on  the  steam  side  to  the  lead :  divide  the 
sum  by  half  the  length  of  stroke  of  the  valve.  In  a  table 
of  natural  sines  find  the  arc  whose  sine  is  equal  to  the  quo- 
tient thus  obtained.  To  this  arc  add  90°,  and  from  the  sum 
of  these  two  arcs  subtract  the  arc  whose  cosine  is  equal  to  the 
lap  on  the  steam  side  divided  by  half  the  stroke  of  the  valve. 
Find  the  cosine  of  the  remaining  arc,  add  1  to  it,  and  mul- 
tiply the  turn  by  half  the  stroke  of  the  piston,  and  the  prod- 
uct is  the  length  of  that  part  of  the  stroke  that  will  be  made 
by  the  piston  before  the  steam  is  cut  off. 

TO  FIND  HOW  MUCH  BEFOEE  THE  END  OF  THE  STEOKE  THE  EX- 
HAUSTION OF  THE  STEAM  IN  FEONT  OF  THE  PISTON  WILL  BE 
OUT  OFF. 

RULE. — To  the  lap  on  the  steam  side  add  the  lead,  and  divide 
the  sum  by  half  the  length  of  the  stroke  of  the  valve.  Find 
the  arc  whose  sine  is  equal  to  the  quotient,  and  add  90°  to  it. 
Divide  the  lap  on  the  exhausting  side  by  half  the  stroke  of 
the  valve,  and  find  the  arc  whose  cosine  is  equal  to  the  quo- 
tient. Subtract  this  arc  from  the  one  last  obtained,  and 
find  the  cosine  of  the  remainder.  Subtract  this  cosine  from 


188  THEORY   OF   THE    STEAM-ENGINE. 

2,  and  multiply  the  remainder  l>y  lialf  the  stroke  of  the  pis- 
ton. The  product  is  the  distance  of  the  piston  from  the  end 
of  the  stroke  when  the  exhaustion  is  cut  off. 


TO  FIND  HOW  FAB  THE  PISTON  IS  FROM  THE  END  OF  ITS  STEOKE, 
WHEN  THE  STEAM  THAT  IS  PROPELLING  IT  BY  EXPANSION  13  AL- 
LOWED TO  ESCAPE  TO  THE  CONDENSER. 

RULE. — To  the  lap  on  the  steam  side  add  the  lead ;  divide  the 
sum  by  half  the  stroke  of  the  valve,  and  find  the  arc  whose 
sine  is  equal  to  the  quotient.  Find  the  arc  whose  cosine  is 
equal  to  the  lap  on  the  exhausting  side,  divided  by  half  the 
stroke  of  the  valve.  Add  these  two  arcs  together,  and  sub- 
tract 90°.  Find  the  cosine  of  the  residue,  subtract  it  from  1, 
and  multiply  the  remainder  by  half  the  stroke  of  the  piston. 
The  product  is  the  distance  of  the  piston  from  the  end  of  its 
stroke,  when  the  steam  that  is  propelling  it  is  allowed  to  es- 
cape to  the  condenser. 

NOTE. — In  using  these  rules  all  the  dimensions  are  to  be  taken 
in  inches,  and  the  answers  will  he  found  in  inches  also. 

It  will  readily  be  perceived  from  a  consideration  of  these 
rules  that — supposing  there  is  no  lead — the  point  of  the  stroke 
at  which  the  steam  is  cut  off  is  determined  by  the  proportion 
which  the  lap  on  the  steam  side  bears  to  the  stroke  of  the  valve. 
Whatever  the  absolute  dimensions  of  the  lap  may  be,  therefore, 
it  will  follow  that,  in  every  case  in  which  it  bears  the  same  ratio 
to  the  stroke  of  the  valve,  the  steam  will  be  cut  off  at  the  same 
point  of  the  stroke. 

As  some  of  the  foregoing  rules  are  difficult  to  be  worked  out 
by  persons  unacquainted  with  trigonometry,  it  will  be  conven- 
ient to  collect  the  principal  results  into  tables,  which  may  be 
applied  without  difficulty  to  the  solution  of  any  particular  ex- 
ample. This  accordingly  has  been  done  in  the  three  following 
tables,  the  mode  of  using  which  it  will  now  be  proper  to  ex- 
plain. 


RELATIONS  OF  LAP  AND  EXPANSION. 


189 


I. PROPORTION    OF   LAP   REQUIRED   TO    ACCOMPLISH   VARIOUS   DEGREES   OP 

EXPANSION. 


Distance  of  the  piston  "I 

from  the  termina- 

21 

j* 

?* 

i'» 

5* 

tion  of  its  stroke, 

when  the  steam  is  V 

or 

5*t 

or 

« 

or 

or 

or 

A 

cut  off,  in  parts  of 

the    length  of    its  | 

i 

i 

i 

8 

A 

stroke    J 

Lap  on  the  steam  side  ~) 

of  the  valve,  In  de-  I 
clmal  parts  of  the  | 

•289 

•270 

•250 

•228 

•204 

•ITT 

•144 

•102 

length  of  its  stroke.  J 

Example. — In  the  first  line  of  the  first  table  will  be  found 
eight  different  parts  of  the  stroke  of  the  piston  designated ;  and 
directly  below  each,  in  the  second  line,  is  given  the  quantity  of 
lap  requisite  to  cause  the  steam  to  be  cut  off  at  that  particular 
part  of  the  stroke.  The  different  amounts  of  the  lap  are  given 
in  the  second  line  in  decimal  parts  of  the  length  of  the  stroke 
of  the  valve ;  so  that,  to  get  the  quantity  of  lap  corresponding 
to  any  of  the  given  degrees  of  expansion,  it  is  only  necessary  to 
take  the  decimal  in  the  second  line,  which  stands  under  the  frac- 
tion in  the  first,  that  marks  that  degree  of  expansion,  and  mul- 
tiply that  decimal  by  the  length  we  intend  to  make  the  stroke 
of  the  valve.  Thus  suppose  we  have  an  engine  in  which  we 
wish  to  have  the  steam  cut  off  when  the  piston  is  a  quarter  of 
the  length  of  its  stroke  from  the  end  of  it,  we  look  in  the  first 
line  of  the  table,  and  we  shall  find  in  the  third  column  from  the 
left,  \.  Directly  under  that,  in  the  second  line,  we  have  the 
decimal,  '250.  Suppose  that  we  consider  that  18  inches  will  be 
a  convenient  length  for  the  stroke  of  the  valve,  we  multiply  the 
decimal  -250  by  18,  which  gives  4J.  Hence  we  learn,  that  with 
an  18-inch  stroke  for  the  valve,  4£  inches  of  lap  on  the  steam 
side  will  cause  the  steam  to  be  cut  off  when  the  piston  has  still 
a  quarter  of  its  stroke  to  perform. 

Half  the  stroke  of  the  valve  should  always  be  at  least  equal 
to  the  lap  on  the  steam  side  added  to  the  breadth  *  of  the  port ; 
consequently,  as  the  lap  in  this  case  must  be  4£  inches,  and  as 

*  By  the  '  breadth '  of  the  port,  is  meant  its  dimensions  in  the  direction  of  the. 
valro's  motion ;  in  short,  its  perpendicular  depth  when  the  cylinder  is  upright 


190 


THEORY    OF   THE    STEAM-ENGINE. 


half  the  stroke  of  the  valve  is  9  inches,  the  efficient  breadth  of 
the  port  cannot  he  more  than  9  —  4£  =  44-  inches,  since  half  of  it 
is  covered  over  hy  the  lap.  If  this  breadth  of  port  is  not  suffi- 
cient to  give  the  required  area  to  let  the  steam  in  and  out,  we 
must  increase  the  stroke  of  the  valve ;  by  which  means  we  shall 
get  both  the  lap  and  the  breadth  of  the  port  proportionally  in- 
creased. Thus,  if  we  make  the  length  of  valve-stroke  20  inches, 
we  shall  have  for  the  lap  '250  x  20=5  inches,  and  for  the  breadth 
of  the  port  10  —  5  =  5  inches. 

This  table,  as  we  have  already  intimated,  is  computed  on  the 
supposition  that  the  valve  is  to  have  no  lead ;  but,  if  it  is  to 
have  lead,  all  that  is  necessary  is  to  subtract  half  the  proposed 
lead  from  the  lap  found  from  the  table,  and  the  remainder  will 
be  the  proper  quantity  of  lap  to  give  to  the  valve.  Suppose 
that,  in  the  last  example,  the  valve  was  to  have  J  inch  of  lead, 
wo  should  subtract  £  inch  from  the  5  inches,  found  for  the  lap 
by  the  table.  This  would  leave  4$  inches  for  the  quantity  of 
lap  that  the  valve  ought  to  have. 

n. LAP   IN    INCHES   REQUIRED   ON   THE    STEAM   SIDE   OF   THE  VALVE   TO  CUT 

THE   STEAM   OFF   AT   ANY   OF   THE   UNDER-NOTED   PARTS   OF   THE    STROKE. 


Length 
of  stroke 
of  the 
valve  in 
inches. 

Proportion  of  the  stroke  at  which  the  steam  is  cut  off. 

3 

JL 

t 

ft 

i 

i 

A 

A 

24 

6-94 

6-48 

6-00 

5-47 

4-90 

4-25 

3-47 

2-45 

23* 

6-79 

6-34 

5-88 

5-36 

4-79 

4-16 

3-39 

2-39 

23 

6-65 

6-21 

5-75 

5-24 

4-69 

4-07 

3-32 

2-34 

22* 

6-50 

6-07 

5-62 

5-13 

4-59 

3-98 

8-25 

2-29 

22 

6-36 

5-94 

5-50 

5-02 

4-49 

3-89 

3-13 

2-24 

21* 

6-21 

5-80 

5-38 

4-90 

4-39 

3-80 

3-10 

2-19 

21 

6-07 

5-67 

5-25 

4-79 

4-28 

3-72 

3-03 

2-14 

20* 

5-92 

5-53 

5-1.2 

4-67 

4-18 

3-63 

2-96 

2-09 

20 

5-78 

5-40 

5-00 

4-56 

4-08 

3-54 

2-89 

2-04 

19* 

5-64 

5-26 

4-87 

4-45 

3-98 

3-45 

2-82 

1-99 

19 

5-49 

5-13 

4-75 

4-33 

3-88 

3-36 

2-74 

1-94 

18* 

5-34 

4-99 

4-62 

4-22 

3-77 

3-27 

2-67 

1-88 

18 

5-20 

4-86 

4-50 

4-10 

3-67 

3-19 

2-60 

1-83 

17* 

5-06 

4-72 

4-37 

3-99 

3-57 

3-10 

2-53 

1-78 

17 

4-91 

4-59 

4-25 

3-88 

3-47 

3-01 

2-45 

1-73 

16* 

4-77 

4-45 

4-12 

3-76 

3-36 

2-92 

2-38 

1-68 

16 

4-62 

4-32 

4-00 

3-65 

3-26 

2-83 

2-31 

1-G3 

15* 

4-48 

4-18 

3-87 

3-53 

3-16 

2-74 

2-24 

1-58 

PROPORTIONS    OF   LAP   FOR   EXPANSION. 
TABLE —  Continued. 


191 


Length 
of  stroke 
of  the 
valve  in 
inches. 

Proportion  of  the  stroke  at  which  the  steam  is  cut  off. 

a 

-h 

4 

A 

* 

i 

A 

& 

15 

4-33 

4-05 

3-75 

3-42 

3-06 

2-65 

2-16 

1-53 

14* 

4-19 

3-91 

3-62 

3-31 

2-96 

2-57 

2-09 

1.-48 

14 

4-05 

3-78 

3-50 

3-19 

2-86 

2-48 

2-02 

1-43 

13^ 

3-90 

3-64 

3-37 

3-03 

2-75 

2-39 

1-95 

1-37 

13 

3-76 

3-51 

3-25 

2-96 

2-65 

2-30 

1-88 

1-32 

12* 

3-61 

3-37 

3-12 

2-85 

2-55 

2-21 

1-80 

1-27 

12 

3-47 

3-24 

3-00 

2-74 

245 

2-12 

1-73 

1-22 

11* 

3-32 

310 

2-87 

2-62 

2-35 

2;03 

1-66 

1-17 

11 

3-18 

2-97 

2-75 

2-51 

2-24 

1-95 

1-58 

1-12 

10* 

3-03 

2-83 

2-62 

2-39 

2-14 

1-86 

1-51 

1-07 

10 

2-89 

2-70 

2-50 

2-28 

2-04 

1-77 

1-44 

1-02 

9* 

2-65 

2-56 

2-37 

2-17 

1-93 

1-68 

1-32 

•96 

9 

2-60 

2-43 

2-25 

2-05 

1-84 

1-59 

1-30 

•92 

8* 

2-46 

2-29 

2-12 

1-94 

1-73 

1-50 

1-23 

•86 

8 

2-31 

2-16 

2-00 

1-82 

1-63 

1-42 

1-15 

•81 

H 

216 

2-02 

1-87 

1-71 

1-53 

1-33 

1-08 

•76 

7 

2-02 

1-89 

1-75 

1-60 

1-43 

1-24 

1-01 

•71 

H 

1-88 

1-75 

1-62 

1-48 

1-32 

1-15 

•94 

•66 

6 

1-73 

1-62 

1-50 

1-37 

1-22 

1-06 

•86 

•61 

H 

1-58 

1-48 

1-37 

1.25 

1-12 

•97 

•79 

•56 

5 

1-44 

1-35 

1-25 

1-14 

1-02 

•88 

•72 

•51 

** 

1-30 

1-21 

1-12 

1-03 

•92 

•80 

•65 

•46 

4 

1-16 

1-08 

1-00 

•91 

•82 

•71 

•58 

•41 

*t 

1-01 

•94 

•87 

•80 

•71 

•62 

•50 

•35 

3 

•86 

•81 

•75 

•68 

•61 

•53 

•44 

•30 

The  above  table  is  an  extension  of  tho  first,  for  the  purpose 
of  obviating,  in  most  cases,  the  necessity  of  even  the  very  small 
degree  of  trouble  required  in  multiplying  the  stroke  of  the  valve 
by  one  of  the  decimals  in  the  first  table.  The  first  line  of  the 
second  table  consists,  as  in  the  first  table,  of  eight  fractions,  in- 
dicating tho  various  parts  of  the  stroke  at.  which  the  steam  may 
be  cut  off.  The  first  column  on  the  left  hand  consists  of  various 
numbers  that  represent  the  different  lengths  that  may  be  given 
to  the  stroke  of  the  valve,  diminishing  by  half  inches  from  24 
inches  to  3  inches.  Suppose  that  we  wish  the  steam  to  be  cut 


192  THEORY   OF   THE    STEAM-ENGINE. 

off  at  any  of  the  eight  parts  of  the  stroke  indicated  in  the  first 
line  of  the  table  (say  at  %  from  the  end  of  the  stroke),  we  find 
%  at  the  top  of  the  6th  column  from  the  left.  "We  next  look  for 
the  proposed  length  of  stroke  of  the  valve  (say  17  inches)  in  the 
first  column  on  the  left.  From  17,  in  that  column,  we  run  along 
the  line  towards  the  right,  and  in  the  sixth  column,  and  directly 
under  the  %  at  the  top,  we  find  3 '47,  which  is  the  amount  of  lap 
required  in  inches  to  cause  the  steam  to  be  cut  off  at  £  from  the 
end  of  the  stroke,  if  the  valve  has  no  lead.  If  we  wish  to  give 
it  lead  (say  J  inch),  we  subtract  the  half  of  that,  or  £='125  inch, 
from  3-47,  and  we  have  3'47—125=3'345  inches,  the  quantity 
of  lap  that  the  valve  should  have. 

To  find  the  greatest  efficient  breadth  that  we  can  give  to 
the  port  in  this  case,  we  have,  as  before,  half  the  length  of 
stroke,  8£— 3'345=5'155  inches,  which  is  the  greatest  efficient 
breadth  we  can  give  to  the  port  with  this  length  of  stroke.  It 
is  scarcely  necessary  to  observe  that  it  is  not  at  all  essential  that 
the  port  should  be  so  broad  as  this;  indeed,  where  great  length 
of  stroke  in  the  valve  is  not  inconvenient,  it  is  always  an  advan- 
tage to  make  it  travel  further  than  is  just  necessary  to  make  the 
port  open  fully ;  because,  when  it  travels  further,  both  the  ex- 
hausting and  steam  ports  are  more  quickly  opened,  so  as  to  al- 
low greater  freedom  of  motion  to  the  steam. 

The  manner  of  using  this  table  is  so  simple,  that  we  need 
not  trouble  ourselves  with  more  examples,  and  may  pass  on, 
therefore,  to  explain  the  use  of  the  third  table. 

Suppose  that  the  piston  of  a  steam-engine  is  making  its 
downward  stroke,  that  the  steam  is  entering  the  upper  part  of 
the  cylinder  by  the  upper  steam  port,  and  escaping  from  below 
the  piston  by  the  lower  exhausting  port ;  if,  as  is  generally  the 
case,  the  slide-valve  has  some  lap  on  the  steam  side,  the  upper 
port  will  be  closed  before  the  piston  gets  to  the  bottom  of  the 
stroke,  and  the  steam  above  then  acts  expansively,  while  the 
communication  between  the  bottom  of  the  cylinder  and  the  con- 
denser still  continues  open,  to  allow  any  vapour  from  the  con- 
densed water  in  the  cylinder,  or  any  leakage  past  the  piston,  to 
escape  into  the  condenser ;  but,  before  the  piston  gets  to  the 


EFFECTS    OF    LAP    ON   EDUCTION.  193 

bottom  of  the  cylinder,  this  passage  to  the  condenser  will  also  be 
cut  off  by  the  valve  closing  the  lower  port.  Soon  after  the  lower 
port  is  thus  closed,  the  upper  port  will  be  opened  towards  the 
condenser,  so  as  to  allow  the  steam  that  has  been  acting  expan- 
sively to  escape.  Thus,  before  the  piston  has  completed  its 
stroke,  the  propelling  power  is  removed  from  behind  it,  and  a 
resisting  power  is  opposed  before  it,  arising  from  the  vapour  in 
the  cylinder,  which  has  no  longer  any  passage  open  to  the  con- 
denser. It  is  evident,  that  if  there  is  no  lap  on  the  exhausting 
side  of  the  valve,  the  exhausting  port  before  the  piston  will  be 
closed,  and  the  one  behind  it  opened,  at  the  same  time ;  but,  if 
there  is  any  lap  on  the  exhausting  side,  the  port  before  the  pis- 
ton will  be  closed  before  that  behind  it  is  opened ;  and  the  in- 
terval between  the  closing  of  the  one  and  the  opening  of  the 
other,  will  depend  on  the  quantity  of  lap  on  the  exhausting  side 
of  the  valve.  Again,  the  position  of  the  piston  in  the  cylinder, 
when  these  ports  are  closed  and  opened  respectively,  will  depend 
on  the  quantity  of  lap  that  the  valve  has  on  the  steam  side.  If 
the  lap  is  large  enough  to  cut  the  steam  off  when  the  piston  is 
yet  a  considerable  distance  from  the  end  of  its  stroke,  these 
ports  will  be  closed  and  opened  at  a  proportionably  early 
part  of  the  stroke ;  and  in  the  case  of  engines  moving  at 
a  high  speed,  it  has  been  found  that  great  benefit  is  obtained 
from  allowing  the  steam  to  escape  before  the  end  of  the 
stroke. 

The  third  table  is  intended  to  show  the  parts  of  the  stroke 
where,  under  any  given  arrangement  of  slide  valve,  the  eduction 
ports  close  and  open  respectively,  so  that  thereby  the  engineer 
may  be  able  to  estimate  how  much,  if  any,  of  the  efficiency  he 
loses,  while  he  is  trying  to  add  to  the  power  of  the  steam  by  in- 
creasing the  expansion.  In  this  table  there  are  eight  columns 
marked  A,  standing  over  eight  columns  marked  B,  and  at  the 
heads  of  these  columns  are  eight  fractions  as  before,  representing 
so  many  different  parts  of  the  stroke  at  which  the  steam  may 
be  supposed  to  be  cut  off. 


194 


THEOBY    OF    THE    STEAM-ENGINE. 


The  columns  marked  A  express  the  distance  of  the  piston— 
in  parts  of  its  stroke— from  the  end  of  the  stroke  when  the  educ- 
tion port  before  it  is  shut,  and  the  columns  marked  B,  and  which 
stand  immediately  under  the  columns  marked  A,  express  the 
distance  of  the  piston  from  the  end  of  its  stroke  when  the  ex- 
hausting port  behind  it  is  opened — also  measured  in  parts  of  the 
stroke.* 

III. PBOPOETION   OF   THE    STROKE   AT   WHICH   THE   EDUCTION 

POBT  IS   SHUT   AND    OPENED. 


Lap  on  the 
•ductLn  aide  of  the 
valve,  in  parts  of 
the  length  of  its 
stroke. 

Proportion  of  the  stroke  at  which  the  steam  is  cut  off. 

i 

A 

i 

A 

i 

? 

A 

A 

l-8th 
1-1  6th 
l-32nd 
0 

A 

•178 
•130 
•113 
•092 

A 

•161 
•118 
•101 
•082 

A 

•143 
•100 
•085 
•067 

A 

•126 
•085 
•069 
•055 

A 
•109 
•071 
•053 
•043 

A 

•093 
•058 
•043 
•088 

A 

•074 
•043 
•033 
•022 

A 

•053 
•027 
•024 
•Oil 

l-8th 
l-16th 
l-32nd 
0 

B 

•033 
•060 
•073 
•092 

B 

•026 
•052 
•066 
•082 

B 

•019 
•040 
•051 
•067 

B 

•012 
•030 
•042 
•055 

B 

•008 
•022 
•033 
•044 

B 

•004 
•015 
•023 
•033 

B 

•001 
•008 
•013 
•022 

B 

•001 
•002 
•004 

•on 

Suppose  we  have  an  engine  in  which  the  slide  valve  is  made 
to  cut  the  stem  off  when  the  piston  is  l-3rd  from  the  end  of  its 

*  In  locomotive  and  other  fast-moving  engines  it  is  very  important  to  open  the 
eduction  passage  before  the  end  of  the  stroke,  so  as  to  give  more  time  for  the 
steam  to  escape,  and  in  locomotive  valves  the  lap  of  the  valve  is  usually  made  a 
little  over  Jth  of  the  travel,  and  the  lead  is  usually  made  ^th  of  the  travel.  In 
engines  moving  slowly  the  same  necessity  for  an  early  eduction  does  not  exist,  and 
in  such  engines  there  will  be  a  loss  from  opening  the  eduction  much  before  the  end 
of  the  stroke,  as  the  moving  pressure  urging  the  piston  is  thus  removed  before  the 
stroke  tenninates.  "When  the  valve  is  closed  before  the  piston  previously  to  the 
end  of  the  stroke,  the  attenuated  vapour  in  the  cylinder  will  be  compressed,  and 
sometimes  the  compression  will  be  carried  so  far  that  the  pressure  resisting  the 
piston  at  the  end  of  the  stroke  will  exceed  the  pressure  of  the  steam  in  the  boiler. 
The  indicator  diagram  will  in  such  cases  appear  with  a  loop  at  its  upper  corner, 
which  shows  that  the  pressure  before  the  end  of  the  stroke  exceeds  the  pressure  of 
the  steam,  and  that  the  first  effect  of  opening  the  communication  between  the 
cylinder  and  the  boiler  is  to  enable  the  cylinder  to  discharge  its  highly  compressed 
vapour  backward  into  the  boiler.  The  act  of  compressing  the  steam  is  what  is 
called  cushioning  ;  and  in  all  ordinary  diagrams  this  action  may  bo  more  or  less 
perceived. 


EFFECTS    OF   LAP   ON    EDUCATION.  195 

stroke,  and  that  the  lap  on  the  eduction  or  exhausting  side  of 
the  valve  is  l-8th  of  the  whole  length  of  its  stroke.  Let  the 
stroke  of  the  piston  be  6  feet,  or  72  inches.  "We  wish  to  know 
when  the  exhausting  port  before  the  piston  will  be  closed,  and 
when  the  one  behind  it  will  be  opened.  At  the  top  of  the  left- 
hand  column  marked  A,  the  given  degree  of  expansion  (l-3rd) 
is  given,  and  in  the  extreme  left  column  we  have  at  the  top  the 
given  amount  of  lap  (l-8th).  Opposite  the  l-8th  in  the  first 
column,  marked  A,  we  have  '178,  and  in  the  first  column,  marked 
B,  '033,  which  decimals,  multiplied  respectively  by  72,  the  length 
of  the  stroke,  will  give  the  required  positions  of  the  piston: 
thus  72  x  '178  =  12'8  inches  =  distance  of  the  piston  from  the 
end  of  the  stroke  when  the  exhaustion-port  before  the  piston  is 
shut :  and  72  x  '033  =  2'38  inches  =  distance  of  the  piston  from 
the  end  of  its  stroke  when  the  exhausting-port  behind  it  is 
opened. 

To  take  another  example.  Let  the  stroke  of  the  valve  be  16 
inches,  the  lap  on  the  exhausting  side  £  inch,  the  lap  on  the 
steam  side  3J  inches,  and  the  length  of  the  stroke  of  the  piston 
60  inches.  It  is  required  to  ascertain  all  the  particulars  of  the 
working  of  this  valve.  The  lap  on  the  exhausting  side  is  evi- 
dently 5V  of  the  length  of  the  valve  stroke.  Then,  looking  at  16 
in  the  left-hand  column  of  the  table  in  page  190,  we  find  in  the 
same  horizontal  line,  3'26,  or  very  nearly  3J,  under  \  at  the  head 
of  the  column,  thus  showing  that  the  steam  will  be  cut  off  at 
one-sixth  from  the  end  of  the  stroke.  Again,  under  %  at  the 
head  of  the  sixth  column  from  the  left  in  the  table  in  page  194, 
and  in  a  line  with  ^  m  *ne  left-hand  column,  we  have  '053  un- 
der A,  and  -033  under  B.  Hence,  -053  x  60  =  3-18  inches  =  dis- 
tance of  the  piston  from  the  end  of  its  stroke  when  the  exhaust- 
ing-port before  it  is  shut,  and  '033  x  60  =  1'98  inches  =  distance 
of  the  piston  from  the  end  of  its  stroke  when  the  exhausting- 
port  behind  it  is  opened.  If  in  this  valve  the  lap  on  the  ex- 
hausting side  were  increased  say  to  2  inches  or  |  of  the  stroke, 
the  effect  would  be  to  cause  the  port  before  the  valve  to  be  shut 
sooner  in  the  proportion  of  -109  to  '053,  and  the  port  behind  it 
later  in  the  proportion  of  -008  to  -003.  Whereas,  if  the  lap  on 


196  THEORY    OF   THE    STEAM-ENGINE. 

the  exhausting  side  were  removed  entirely,  the  port  before  the 
piston  would  be  shut  and  that  behind  it  opened  at  the  same 
time.  The  distance  of  the  piston  from  the  end  of  its  stroke  at 
that  time  would  be  '043  x  60  =  2'58  inches. 

An  inspection  of  the  third  table  shows  us  the  effect  of  in- 
creasing the  expansion  by  the  slide  valve  in  augmenting  the  loss 
of  power  occasioned  by  the  imperfect  action  of  the  eduction  pas- 
sages. Eeferring  to  the  bottom  line  of  the  table,  we  see  that  the 
eduction  passage  before  the  piston  is  closed,  and  that  behind  i' 
opened,  thus  destroying  the  whole  moving  power  of  the  engine, 
when  the  piston  is  '092  from  the  end  of  its  stroke,  the  steam 
being  cut  off  at  -^  from  the  end.  "Whereas  if  the  steam  is  only 
cut  off  at  ^Y  from  the  end  of  the  stroke,  the  moving  power  is  not 
withdrawn  till  only  -Oil  of  the  stroke  remains  uncompleted.  It 
will  also  be  observed  that  increasing  the  lap  on  the  exhausting 
side  has  the  effect  of  retaining  the  action  of  the  steam  longer 
behind  the  piston,  but  it  at  the  same  time  causes  the  eduction 
port  before  it  to  be  closed  sooner. 

A  very  cursory  examination  of  the  action  of  the  slide  valve 
is  sufficient  to  show  that  the  lap  on  the  steam  side  should  always 
be  greater  than  on  the  eduction  side.  If  they  were  equal,  the 
steam  would  be  admitted  on  one  side  of  the  piston  at  the  same 
time  that  it  was  allowed  to  escape  from  the  other ;  but  universal 
experience  has  shown  that  when  this  is  the  case  a  very  con- 
siderable part  of  the  power  of  the  engine  is  destroyed  by  the  re- 
sistance opposed  to  the  piston,  by  the  escaping  steam  not  getting 
away  to  the  condenser  with  sufficient  rapidity.  Hence  we  see 
the  necessity  of  the  lap  on  the  eduction  side  being  always  less 
than  the  lap  on  the  steam  side  ;  and  the  difference  should  be  the 
greater  the  higher  the  velocity  of  the  piston  is  intended  to  be, 
because  the  quicker  the  piston  moves,  the  passage  for  the  waste 
steam  requires  to  be  the  larger,  so  as  to  admit  of  its  getting  away 
to  the  condenser  with  as  great  rapidity  as  possible.  In  locomo- 
tive or  other  engines,  where  it  is  not  wished  to  expand  the  steam 
in  the  cylinder  at  all,  the  slide  valve  is  sometimes  made  with 
very  little  lap  on  the  steam  side ;  and  in  these  circumstances,  in 
order  to  get  a  sufficient  difference  between  the  lap  on  the  steam 


EFFECTS  OF  LAP  ON  EDUCTION.  197 

and  the  eduction  sides  of  the  valve,  it  may  be  necessary  not  only 
to  take  away  all  the  lap  on  the  eduction  side,  but  to  take  off  still 
more,  so  as  to  cause  both  eduction  passages  to  be,  in  some  de- 
gree, open,  when  the  valve  is  at  the  middle  of  its  stroke.  This, 
accordingly,  is  sometimes  done  in  such  circumstances  as  we  have 
described ;  but,  when  there  is  a  considerable  amount  of  lap  on 
the  steam  side,  this  plan  of  taking  more  than  all  the  lap  off  the 
eduction  side  ought  never  to  be  resorted  to,  as  it  can  serve  no 
good  purpose,  and  will  materially  increase  an  evil  we  have  al- 
ready explained :  viz.,  the  opening  of  the  eduction  port  behind 
the  piston  before  the  stroke  is  nearly  completed.  In  the  case 
of  locomotive  or  other  engines  moving  rapidly,  it  is  very  con- 
ducive to  efficiency  to  begin  the  eduction  before  the  end  of  the 
stroke,  as  the  piston  moves  slowly  at  that  time ;  and  a  very  small 
amount  of  travel  in  the  piston  at  that  point  corresponds  to  a 
considerable  additional  time  given  for  the  accomplishment  of  the 
eduction.  The  tables  apply  equally  to  the  common  short-slide 
three-ported  valves,  and  to  the  long  D  valves. 

The  extent  to  which  expansion  can  be  carried  conveniently 
by  means  of  lap  upon  the  valve  is  about  one-third  of  the  stroke  ; 
that  is,  the  valve  may  be  made  with  so  much  lap  that  the  steam 
will  be  cut  off  when  one-third  of  the  stroke  has  been  performed, 
leaving  the  residue  to  be  accomplished  by  the  agency  of  the  ex- 
panding steam ;  but  if  much  more  lap  be  put  on  than  answers  to 
this  amount  of  expansion  a  distorted  action  of  the  valve  will  be 
produced,  which  will  impair  the  efficiency  of  the  engine.  By 
the  use  of  the  link  motion,  however,  much  of  this  distorted  action 
can  be  compensated.  If  a  farther  amount  of  expansion  than  this 
is  wanted,  where  the  link  motion  is  not  used,  it  may  be  attained 
by  wire-drawing  the  steam,  or  by  so  contracting  the  steam  pas- 
sage that  the  pressure  within  the  cylinder  must  decline  when  the 
speed  of  the  piston  is  accelerated,  as  it  is  about  the  middle  of  the 
stroke.  Thus,  for  example,  if  the  valve  be  so  made  as  to  shut 
off  the  steam  by  the  time  two-thirds  of  the  stroke  have  been 
performed,  and  the  steam  be  at  the  same  time  throttled  in  the 
Bteam  pipe,  the  full  pressure  of  the  steam  within  the  cylinder 
cannot  be  maintained  except  near  the  beginning  of  the  stroke, 


198  THEORY   OF   THE    STEAM-ENGINE. 

where  the  piston  travels  slowly ;  for  as  the  speed  of  the  piston 
increases,  the  pressure  necessarily  subsides,  until  the  piston  ap- 
proaches the  other  end  of  the  cylinder,  where  the  pressure  would 
rise  again  but  that  the  operation  of  the  lap  on  the  valve  by  this 
time  has  had  the  effect  of  closing  the  communication  between 
the  cylinder  and  steam  pipe,  so  as  to  prevent  more  steam  from 
entering.  By  throttling  the  steam,  therefore,  in  the  manner 
here  indicated,  the  amount  of  expansion  due  to  the  lap  may  be 
doubled,  so  that  an  engine  with  lap  enough  upon  the  valve  to 
cut  off  the  steam  at  two-thirds  of  the  stroke,  may,  by  the  aid 
of  wire-drawing,  be  virtually  rendered  capable  of  cutting  off  the 
steam  at  one-third  of  the  stroke. 

The  Link  Motion. — The  rules  and  proportions  here  given, 
are  equally  applicable,  whether  the  valve  is  moved  by  a  single 
eccentric,  or  by  the  arrangement  called  the  link  motion,  and 
which  has  now  been  very  generally  introduced  into  steam  en- 
gines. In  the  link  motion  there  are  two  eccentrics,  one  of 
which  is  set  so  as  to  drive  the  engine  in  one  direction,  and  the 
other  is  set  so  as  to  drive  the  engine  in  the  opposite  direction, 
and  when  the  stud  in  communication  with  the  valve  is  shifted  to 
one  end  of  the  link,  that  stud  partakes  of  the  motion  of  the  for- 
ward eccentric,  whereas,  when  it  is  placed  at  the  other  end  of 
the  link,  it  partakes  of  the  motion  of  the  backing  eccentric.  A 
common  length  of  the  link  is  three  times  the  stroke  of  the  valve. 
Generally  the  stud  is  placed  either  at  one  end  of  the  link  or  the 
other,  not  by  moving  the  stud  but  by  moving  up  or  down  the 
link ;  and  it  is  better  that  this  movement  should  be  vertical,  and 
be  made  by  means  of  a  screw,  than  that  the  movement  should 
be  produced  by  a  lever  travelling  through  an  arc.  The  point 
of  suspension  should  be  near  the  middle  of  the  link  where  its 
motion  is  the  least.  The  link  connects  together  the  ends  of  the 
two  eccentric  rods,  and  is  sometimes  made  straight,  but  gen- 
erally curved,  the  curvature  being  an  arc  of  such  radius  that  the 
link  may  be  raised  up  or  down  without  sensibly  altering  the 
position  of  the  stud  with  which  the  valve  is  connected.  But  the 
link  should  be  convex  or  concave  towards  the  valve,  according 
as  the  eccentric  rods  are  crossed  or  uncrossed  when  the  throw 


VELOCITY   OF   RUNNING   WATER   IN    CONDUITS.         199 

of  the  eccentrics  are  turned  towards  the  link.  In  the  case  of 
new  arrangements  of  engine,  it  is  advisable  to  make  a  skeleton 
model  in  paper  of  the  link  and  its  connexions,  so  as  to  obtain  full 
assurance  that  it  works  in  the  best  way. 


VELOCITY    OF    WATER    IN    RIVERS,   CANALS,   AND    PIPES, 
ANSWERABLE   TO    ANY    GIVEN   DECLIVITY. 

When  a  river  runs  in  its  bed  with  a  uniform  velocity,  the 
gravitation  of  the  water  down  the  inclined  plane  of  the  bed,  is 
just  balanced  by  the  friction.  In  the  case  of  canals,  culverts, 
and  pipes,  precisely  the  same  action  takes  place.  The  head  of 
water,  therefore,  which  urges  the  flow  through  a  pipe,  may  be 
divided  into  two  parts,  of  which  one  part  is  expended  in  giving 
to  the  water  its  velocity,  and  the  other  part  is  expended  in  over- 
coming the  friction.  If  water  be  let  down  an  inclined  shoot,  its 
motion  at  the  top  will  be  slow,  but  will  go  on  accelerating  until 
the  friction  generated  by  the  high  velocity  will  just  balance  the 
gravitation  down  the  plane,  and  after  this  point  has  been  attained, 
the  shoot  may  be  made  longer  and  longer  without  any  increase 
in  the  velocity  of  the  water  taking  place.  In  the  case  of  a  ball 
falling  in  the  air  or  in  water,  the  velocity  of  the  descent  will  go 
on  increasing  until  the  resistance  becomes  so  great  as  to  balance 
the  weight ;  and,  in  the  case  of  a  steam  vessel  propelled  through 
the  water,  the  speed  will  go  on  increasing  until  the  resistance 
just  balances  the  tractive  force  exerted  by  the  engines,  when  the 
speed  of  the  vessel  will  become  uniform.  In  all  these  cases  the 
resistance  increases  with  the  speed ;  and  as  the  speed  increases, 
the  resistance  increases  also,  until  it  becomes  equal  to  the  ac- 
celerating force. 

The  resistance  which  is  occasioned  by  the  friction  of  water 
increases  more  rapidly  than  the  increase  of  the  velocity.  In 
other  words,  there  will  be  more  than  twice  the  friction  with 
twice  the  velocity.  It  is  found  by  experiment  that  the  friction 
of  water  increases  nearly  as  the  square  of  its  velocity,  so  that 
there  will  be  about  four  times  the  resistance  with  twice  the 
speed.  This  law,  however,  is  only  approximately  correct.  The 


200  THEORY   OF   THE    STEAM-ENGINE. 

friction  does  not  increase  quite  so  rapidly  at  high  velocities  as 
the  square  of  the  speed. 

It  is  easy  to  determine  the  friction  in  Ihs.  per  square  foot  of 
any  given  pipe  or  conduit,  with  any  given  velocity  of  the  stream, 
when  the  slope  or  declivity  of  the  surface  of  the  water  is  known. 
For  as  the  gravitation  down  the  inclined  plane  of  the  conduit 
just  balances  the  friction,  the  friction  in  the  whole  length  of  the 
conduit  will  be  equal  to  the  whole  weight  of  the  water  in  it,  re- 
duced in  the  same  proportion  as  any  other  body  descending  an 
inclined  plane.  Thus,  if  the  conduit  be  2,000  feet  long,  and  have 
1  foot  of  fall  in  that  length,  the  total  friction  will  be  equal  to  the 
total  weight  of  the  water  divided  by  2,000,  and  the  friction  per 
square  foot  will  be  equal  to  this  2000th  part  of  the  weight  of  the 
water  divided  by  the  number  of  square  feet  exposed  to  the  water 
in  the  conduit.  The  friction  will  in  all  cases  vary  as  the  rubbing 
surface,  or,  what  is  the  same  thing,  as  the  wetted  perimeter 
As  a  cylindrical  pipe  has  a  less  perimeter  than  any  other  form,  it 
will  occasion  less  resistance  than  any  other  form  to  water  passing 
through  it.  In  like  manner,  a  canal  or  a  ship  with  a  semi- 
circular cross  section  will  have  the  minimum  amount  of  friction. 

The  propelling  power  of  flowing  water  being  gravity,  the 
amount  of  such  power  will  vary  with  the  magnitude  of  the 
stream ;  but  the  resisting  power  being  friction,  which  varies  with 
the  amount  of  surface,  or  in  any  given  length  with  the  wetted 
perimeter,  it  will  follow  that  the  larger  the  area  is  relatively 
with  the  wetted  perimeter,  the  less  will  be  the  resistance  rela- 
tively with  the  propelling  power,  and  the  greater  will  be  the 
velocity  of  the  water  with  any  given  declivity.  Now,  as  the 
circumference  or  perimeter  of  a  pipe  increases  as  the  diameter, 
and  the  area  as  the  square  of  the  diameter,  it  is  clear,  that  with 
any  given  head,  water  will  run  more  swiftly  through  large  pipes 
than  through  small ;  and  in  like  manner  with  any  given  propor- 
tion of  power  to  sectional  area,  large  vessels  will  pass  more 
swiftly  than  small  vessels  through  the  water.  The  sectional 
area  of  a  pipe  or  canal  divided  by  the  wetted  perimeter,  is  what 
is  termed  the  hydraulic  mean  depth,  and  this  depth  is  what 
would  result  if  we  suppose  the  perimeter  to  be  bent  out  to  a 


VELOCITY  OP   RUNNING   WATER   IN   CONDUITS.        201 

straight  line,  and  the  sectional  area  to  be  spread  evenly  over  it, 
so  that  each  foot  of  the  perimeter  had  its  proper  share  of  sec- 
tional area  above  it.  The  greater  the  hydraulic  mean  depth,  the 
greater  with  any  given  declivity  will  be  the  velocity  of  the 
stream.  With  any  given  fall,  therefore,  deep  and  large  rivers 
will  run  more  swiftly  than  small  and  shallow  ones.  The  hy- 
draulic mean  depth  of  a  steam  vessel  will  be  the  indicated  power 
divided  by  the  wetted  perimeter  of  the  cross  section. 

TO  DETERMINE  THE  MEAN  VELOCITY  WITH  WHICH  WATER  WILL 
FLOW  THROUGH  CANALS,  ARTERIAL  DRAINS,  OR  PIPES,  RUN- 
NING PARTLY  OR  WHOLLY  FILLED. 

KULE.  —  Multiply  the  hydraulic  mean  depth-  in  feet  ty  twice  the 
fall  in  feet  per  mile  ;  take  the  square  root  of  the  product  and, 
multiply  it  by  55.  The  result  is  the  mean  velocity  of  the 
stream  in  feet  per  minute.  This  again  multiplied  by  the  sec- 
tional area  in  square  feet  gives  the  discharge  in  cubic  feet 
per  minute. 
Example.—  What  is  the  mean  velocity  of  a  river  falling  a  foot 

in  the  mile,  and  of  which  the  mean  hydraulic  depth  is  8  feet  ? 
Here  8  x  2  =  16,  the  square  root  of  which  is  4,  and  this 

multiplied  by  55  =  220,  which  will  be  the  mean  velocity  of  the 

stream  in  feet  per  minute. 

In  cylindrical  pipes  running  full,  the  hydraulic  mean  depth  is 

one-fourth  of  the  diameter.    For  the  hydraulic  mean  depth  being 

the  area  divided  by  the  wetted  perimeter,it  is  —  —    -  =  —  *  •, 

4' 


*  The  surface,  bottom,  and  mean  velocities  of  rivers  have  fixed  relations  to  one 
another.  Thus,  If  the  surface  velocity  in  inches  per  second  be  denoted  by  V,  the 
mean  velocity  will  be  (V  +  0-5)  —  yT  and  the  bottom  velocity  by  (V  +  1)  —  2  V  V. 
With  surface  velocities  therefore  of  4,16,  82,  64,  and  100  inches  per  second,  the  cor- 
responding mean  velocities  will  be  2-5,  12-5,  26-8,  56-5,  and  90'5  inches  per  second, 
and  the  corresponding  bottom  velocities  will  be  1,  9,  21-6,  49,  and  81  inches  per 
second. 

The  common  rule  for  finding  the  number  of  cubic  feet  of  water  delivered  each 
minute  by  a  pipe  of  any  given  diameter  is  as  follows  :  —  Divide  4-72  times  the  square 
root  of  the  fifth  power  of  the  diameter  of  the  pipe  in  inches  by  the  square  root 
of  the  quotient  obtained  by  dividing  the  length  of  the  pipe  in  feet  by  the  head 
of  'water  in  feet.  Hawksley's  rule  for  ascertaining  the  delivery  in  gallons  per 
hour  is  as  folio  ws  -.—Multiply  15  times  theffth  pcncer  of  the  diameter  of  the  pipe 

9* 


202  THEORY    OF   THE    STEAM-ENGINE. 

M.  Prony  has  shown  by  a  comparison  of  a  large  number  of 
experiments  that  if  H  be  the  head  in  feet  per  mile  required  to 
balance  the  friction,  V  the  velocity  of  the  water  through  the  pipe 
in  feet  per  second,  and  D  the  diameter  of  the  pipe  in  feet,  then 

_  2-25V'2 

T~- 

This  equation  is  identical  with  that  which  has  been  used  by 
Boulton  and  "Watt  in  their  practice  for  the  last  half  century,  and 
which  is  as  follows : — 

If  I  be  the  length  of  the  main  in  miles,  V  the  velocity  of  the 
water  in  the  main  in  feet  per  second,  D  the  diameter  of  the  pipe 
in  feet,  and  2 -25  a  constant, 

2-25ZV2 
then  — pj —  =  feet  of  head  due  to  friction. 

This  equation  put  into  words  gives  us  the  following  Kule: — 

TO  DETERMINE  THE  HEAD  OF  WATER  THAT  WILL  BALANCE  THE 
FRICTION  OF  WATER  RUNNING  WITH  ANT  GIVEN  VELOCITY 
THROUGH  A  PIPE  OF  A  GIVEN  LENGTH  AND  DIAMETER. 

KULE. — Multiply  2'25  times  the  length  of  the  pipe  in  miles  Tjy 
the  square  of  the  velocity  of  the  water  in  the  pipe  in  feet  per 
second,  and  divide  the  product  hy  the  diameter  of  the  pipe  in 
feet.  The  quotient  is  the  head  of  water  in  feet  that  will 
"balance  the  friction. 
The  law  indicated  by  this  Rule  is  expressed  numerically  in 

the  Tables  on  pp.  204,  205. 

in  inches  'by  the  head  of  water  in  feet,  and  divide  the  product  by  the  length  of 
the  pipe  in  yards.  Finally,  extract  the  square  root  of  the  quotient,  which  givet 
the  delivery  in  gallonsper  hour. 

The  annual  rain-fall  in  England  varies  from  20  to  70  inches,  the  mean  being  42 
inches,  and  it  is  reckoned  that  about  T*5ths  of  the  rain-fall  on  any  given  area  may 
be  collected  for  storage.  A  cubic  foot  of  water  is  about  6J  gallons,  and  it  is  found 
in  supplying  towns  with  water  that  about  on  the  average  16  gallons  per  head  per 
day  are  required  in  ordinary  towns,  and  20  gallons  per  head  per  day  in  manufac- 
turing towns,  but  the  pipes  should  be  large  enough  to  convey  twice  this  quantity. 
In  the  rainy  districts  of  England  collecting  reservoirs  should  contain  120  days'  sup- 
ply, and  in  dry  districts  200  days'  supply.  Service  reservoirs  are  usually  made  to 
contain  3  days'  supply.  The  mean  daily  evaporation  in  England  is  '08  of  an  inch, 
and  the  loss  from  the  overflow  of  storm  water  is  reckoned  to  be  about  10  per  cent. 


FRICTIOX    AND    DISCHARGE    OF   WATER.  203 

Explanation  of  the  Tables.— The  top  horizontal  row  of 
figures  represents  either  the  diameter  of  a  cylindrical  pipe,  or 
four  times  the  area  of  any  other  shaped  pipe  divided  by  the  cir- 
cumference, or  four  times  the  area  of  the  cross  section  of  a  canal, 
divided  by  the  sum  of  all  its  sides,  or  bottom  and  sides,  all  being 
in  inches. 

The  first  vertical  column  indicates  the  slope  of  the  pipe  or 
canal,  that  is,  the  whole  length  of  the  pipe  or  canal,  divided  by 
the  perpendicular  fall. 

Any  number  in  any  other  column  indicates  the  velocity,  in 
inches  per  second,  with  which  water  would  run  through  a  pipe 
of  such  a  diameter  as  the  number  at  the  head  of  such  column 
expresses,  having  such  a  slope  as  that  number  in  the  first  column 
expresses  which  is  horizontally  against  such  velocity. 

Example  1. — With  what  velocity  will  water  run  through  a 
pipe  of  16  inches  diameter,  its  length  being  8,000  feet,  and  fall 
16  feet?  Here  the  slope  manifestly  is  8,000-5-16=500.  Against 
600  in  the  first  column,  and  under  16,  the  diameter  in  the  top 
row  of  figures,  the  number  29'8  is  found,  which  is  the  velocity 
in  inches  per  second. 

Example  2. — With  what  velocity  will  water  pass  through  a 
pipe  of  21  inches  diameter,  having  a  slope  of  900  ?  21  is  not 
found  in  the  head  of  the  Table,  in  which  case  such  a  number 
must  be  found  in  the  top  row  as  will  bear  such  proportion  to  21 
as  some  other  two  numbers  in  the  top  row  bear  to  each  other, 
and  these  latter  numbers  should  be  as  near  to  21  as  they  can  be 
found. 

In  this  case  it  will  be  seen  that  18  is  to  21  as  6  is  to  7,  or 
(for  compliance  with  the  indication  just  mentioned)  rather  as  12 
to  14,  or  still  better  as  24  to  28.  Then  say  as  the  velocity 
(against  900,  the  slope)  under  24  is  to  28  (28'7),  so  is  the  velocity 
under  18  (22'7)  to  that  of  21  (viz.  24-7)  the  velocity  in  inches 
per  second. 

By  the  same  process  may  the  velocity  for  slopes  be  found  or 
assigned,  which  are  not  to  be  found  in  the  first  column  of  the 
Table,  proceeding  with  proportions  found  in  the  vertical  col- 
umn instead  of  the  horizontal  rows;  the  first  vertical  column 


204 


THEORY    OF   THE    STEAM-ENGINE. 


VELOCITY   IN  INCHES   PEK   SECOND   OF   WATER  FLOWING   THROUGH 

PIPES    WITH    VARIOUS   SLOPES   AND   DIAMETERS. 

BY  BOCTLTON,   WATT  &  CO. 


Slope  or 
Length 
divided  by 
FalL 

INTERNAL  DIAMETERS  OF  THE  PIPES  IN  INCHES. 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

12 

14 

10 

68-3 

96-0 

121- 

142- 

161- 

180- 

193- 

208- 

221- 

234- 

255- 

280- 

20 

41-8 

63-4 

80- 

93-6 

104- 

118- 

127- 

187- 

146- 

154- 

170- 

185- 

30 

82-8 

49-5 

62-6 

73-4 

83-3 

931 

99-7 

107- 

114- 

121- 

133- 

145- 

40 

27-5 

41-5 

52-5 

61-5 

69-8 

78-0 

83-7 

901 

95'8 

103- 

112- 

121- 

50 

23'9 

36-3 

45'7 

53'7 

60-9 

68*0 

73'0 

787 

83'5 

88-4 

97'5 

106' 

60 

21-6 

82-7 

41-2 

48-3 

54-8 

61-2 

65-7 

70  -8 

75-8 

79-8 

87-8 

95-4 

70 

19-6 

29-7 

87-5 

44-0 

49-8 

55-7 

59-7 

64-4 

68-5 

72-5 

80-0 

86-6 

80 

181 

27-4 

84-7 

40-7 

461 

51-5 

65-8 

59-5 

63-8 

671 

74-0 

80-2 

90 

16-9 

25-6 

32-4 

37-9 

43-0 

48-0 

51-5 

55-5 

59-0 

62-5 

69-0 

74-8 

100 

15'8 

24-0 

30'3 

35'5 

40-2 

45'1 

48'4 

52'2 

55'5 

58'7 

64-8 

70-3 

200 

10-5 

16-0 

20-2 

23-7 

26-8 

80-0 

82-2 

34-7 

86-9 

39-0 

431 

46-7 

300 

8-43 

12-7 

16-1 

18-6 

21-4 

23-8 

25-6 

27-7 

29-4 

80-6 

34-3 

87-7 

400 

7-11 

10-8 

136 

15-9 

181 

20'2 

21-6 

23-3 

24-8 

26-8 

29-0 

81-4 

500 

6-26 

9-50 

12'0 

14'0 

15'9 

17'8 

191 

20-6 

21'9 

23*2 

25'5 

27'7 

600 

5-64 

8-57 

10-8 

12-7 

14-3 

161 

17-2 

IS'6 

19-7 

20-9 

28-0 

25-0 

700 

517 

7-85 

9-90 

11-6 

13-2 

14-7 

15-8 

17-0 

181 

19-2 

211 

22-9 

800 

4-81 

7-30 

9-21 

10-8 

12-2 

13-7 

14-7 

15-8 

16-8 

17-8 

19-6 

21-8 

900 

4-50 

6-83 

8-62 

101 

11-4 

12-8 

18-7 

14-8 

15-7 

16-6 

18-8 

19-9 

1000 

4-25 

6'45 

8'15 

9-54 

10-8 

12'1 

12'9 

14'0 

14-8 

15-7 

17-3 

18'8 

2000 

2-88 

4-37 

6-52 

6-48 

7-83 

8-2 

8-77 

9-48 

101 

10-6 

11-7 

12-7 

3000 

2-30 

8-48 

4-40 

517 

5-86 

6-55 

7-02 

7-57 

8-05 

8-52 

9-40 

10-2 

4000 

1-96 

2-97 

8-75 

4-40 

4-98 

5-57 

5-97 

6-44 

6-84 

7-25 

8-00 

8-66 

5000 

173 

2'62 

3'31 

3-88 

4-40 

4-92 

5  '28 

5-69 

6-04 

6'40 

7'06 

7'66 

6000 

1-57 

2-88 

8-00 

8-52 

8-99 

4-45 

4-79 

515 

5-44 

5-80 

6-40 

6-95 

7000 

1-43 

2-17 

2-78 

8-21 

8-68 

4-06 

4-36 

4-70 

600 

5-29 

6-82 

6-82 

8000 

1-32 

2-01 

2-58 

2-97 

8-46 

8-76 

4-03 

4-35 

4-62 

4-90 

5-40 

5-85 

9000 

1-24 

1-87 

2-38 

2-79 

816 

8-53 

8-79 

4-08 

4-84 

4-60 

5-07 

5-50 

10000 

1'17 

1-77 

2-24 

2-62 

2'9 

3'32 

3'57 

3*84 

4'08 

4'32 

476 

5-16 

being  substituted  in  this  case  for  the  top  row  in  the  former 
case. 

In  all  cases  an  addition  must  be  made  to  the  fall  equal  to  that 
which  would  generate  the  existing  velocity  in  a  body  falling 
freely  by  gravity.  For  instance,  in  the  first  case,  to  the  fall  of 


VELOCITY   OF   WATER   IN    PIPES. 


205 


TELOCITY  IN  INCHES   PEK   SECOND   OP   WATER  FLOWING/  THROUGH 
PIPES  WITH  VARIOUS  SLOPES  AND  DIAMETERS.  —  (Continued.) 

BY  BOUXTON,   WATT  &   CO. 


Slope  or 
Length 
divided  by 
Fall. 

INTERNAL  DIAMETERS  OF  THE  PIPES  IN  INCHES. 

16 

18 

20 

24 

28 

32 

36 

40 

60 

80 

100 

10 

301- 

320- 

338- 

872- 

403- 

432- 

459- 

484- 

597- 

692- 

775- 

20 

199- 

211- 

223- 

245- 

256- 

235- 

303- 

320- 

894- 

456- 

511- 

30 

loo- 

165- 

175- 

192- 

208- 

223- 

238' 

250- 

309- 

358- 

400- 

40 

ISO- 

138- 

wo- 

161- 

174- 

187- 

199- 

210- 

259- 

300- 

836- 

50 

113' 

121' 

rn1 

140' 

152' 

163' 

173' 

183' 

226' 

261' 

293' 

60 

102- 

109- 

115- 

127- 

187- 

147- 

156- 

165- 

204- 

236- 

264- 

70 

93-2 

99-0 

104- 

115- 

124- 

134- 

142- 

150- 

185- 

214- 

240- 

80 

86-8 

91-6 

97- 

106- 

115- 

123- 

131- 

139- 

171- 

198- 

222- 

90 

80-5 

85-5 

90-4 

99-4 

107- 

115- 

122- 

129- 

159- 

185- 

207- 

100 

75'5 

80-2 

85'0 

93-5 

102- 

108' 

115' 

121- 

150' 

173' 

194' 

200 

BO-3 

53-3 

56-4 

62-0 

67-2 

72-1 

76-7 

80-8 

99-5 

115- 

129- 

300 

89-9 

42-5 

45-0 

49-5 

53-6 

57-5 

61-1 

64-4 

79-5 

92-0 

105- 

400 

88-8 

85-9 

38-0 

41-8 

45-2 

4S-5 

51-5 

54-4 

67-0 

77-7 

87-0 

500 

29'8 

31-6 

33'4 

36-8 

39-8 

42-8 

45-5 

47'9 

59'0 

68-4 

767 

600 

26-8 

28-6 

80-2 

88-2 

86-0 

88-6 

41-0 

43-2 

63-3 

61-7 

69-2 

700 

24-6 

28-2 

27-7 

80-4 

83-0 

85-8 

87-6 

39-6 

48-8 

56-5 

68-4 

800 

22-9 

24-3 

25-7 

28-8 

80-6 

82-9 

84-9 

86-S 

45-4 

52-5 

68-9 

900 

21-4 

22-7 

24-0 

26-4 

28-7 

30-7 

32-7 

34-4 

42-5 

49-1 

65-1 

1000 

20'2 

21'5 

227 

25'0 

27'1 

29'1 

30-9 

32'5 

40'1 

46-4 

52-0 

2000 

18-7 

14-5 

15-4 

16-9 

18-8 

19-7 

20-9 

22-0 

27-2 

81-3 

85-8 

3000 

10-9 

11-6 

12-8 

18-5 

14-6 

15-7 

16-7 

17-6 

21-7 

25-2 

28-2 

4000 

9-32 

9-90 

10-4 

11-5 

12-4 

13-4 

14-2 

15-0 

18-5 

21-4 

24-0 

5000 

8-23 

8'73 

9'25 

10-3 

ll'O 

11-8 

12'5 

13'2 

16'3 

18-9 

21'2 

6000 

7-47 

7-98 

8-40 

9-23 

10-0 

10-7 

11-4 

12-0 

14-8 

17-1 

19-2 

7000 

6-80 

7-22 

7-65 

8-40 

9-10 

9-76 

10-8 

10-9 

18-5 

15-6 

17-5 

8000 

6-80 

6-69 

7-07 

7-78 

8-48 

9-05 

9-62 

10-1 

12-8 

14-5 

16-2 

9000 

5-91 

6-28 

6-64 

7-30 

7-92 

8-50 

9-02 

9-62 

11-7 

13-05 

15-2 

10000 

5'55 

5-91 

6-25 

6'87 

7'45 

7'98 

8-50 

8'93 

ll'O 

12'7 

14-3 

16  feet  we  must  add  the  fall  which  would  generate  the  velocity 
of  2  9 '8  inches  per  second,  namely,  1'15  inches,  which  will  make 
the  total  fall  16  feet  1'15  inches  that  will  be  requisite  to  give 
such  a  velocity;  but  in  such  cases  as  this  it  is  evident  that  tho 
addition  of  this  small  fraction  might  have  been  disregarded. 


206  THEOKY    OF    THE    STEAM-ENGINE. 

In  some  cases  Messrs.  Boulton  and  Watt  have  employed  the 
constant  l-82  instead  of  2'25.  Mr.  Mylne's  constant  is  1'94; 
but  some  careful  experiments  made  by  him  at  the  West  Middle- 
sex Waterworks,  gave  a  constant  as  high  as  2-62. 

OTHER    TOPICS    OF    THE    THEOKY    OF    STEAM-ENGINES. 

It  will  not  be  necessary  to  extend  these  remarks  by  an  inves- 
tigation of  the  theory  of  the  crank  as  an  instrument  for  convert- 
ing rectilinear  into  rotatory  motion,  since  the  idea,  once  widely 
prevalent,  that  there  was  a  loss  of  power  consequent  upon  its 
use,  is  now  universally  exploded.  Neither  will  it  be  necessary 
to  enter  into  any  explanation  of  the  structure  of  the  numerous 
rotatory  engines  which  have  at  different  times  been  projected, 
since  none  of  those  engines  are  in  common  or  beneficial  opera- 
tion. The  proper  dimensions  of  the  cold  water  and  feed  pumps, 
the  action  of  the  fly-wheel  in  redressing  irregularities  of  the  mo- 
tion of  the  engine,  and  other  material  points  which  might  prop- 
erly fall  to  be  discussed  under  the  head  of  the  Theory  of  the 
Steam-Engine,  and  which  have  not  already  been  treated  of, 
will,  for  the  sake  of  greater  conciseness,  be  disposed  of  in  the 
chapter  on  the  Proportions  of  Steam-Engines,  when  these  vari- 
ous topics  must  necessarily  be  considered.  Nor  is  it  deemed  ad- 
visable here  to  recapitulate  the  rules  for  proportioning  the  vari- 
ous kinds  of  parallel  motion,  since  parallel  motions  have  now 
almost  gone  out  of  use,  and  since  also  any  particular  case  of  a 
parallel  motion  which  has  to  be  considered,  can  easily  be  resolv- 
ed geometrically  by  drawing  the  parts  on  a  convenient  scale, — 
the  principle  of  all  parallel  motions  being  that  the  versed  sine 
of  an  arc,  pointing  in  one  direction,  shall  be  compensated  by  an 
equal  versed  sine  of  an  arc  pointing  in  the  opposite  direction ;  and 
the  effect  of  these  opposite  motions  is  to  produce  a  straight  line. 
In  the  case  of  the  parallel  motions  sometimes  employed  in  side- 
lever  engines,  and  in  which  the  attachment  is  made  not  to  the 
cross-head  but  to  the  side-rod,  it  is  only  necessary  to  provide 
that  the  end  of  the  bar  connected  to  the  side-rod  shall  move, 
not  in  a  straight  line,  but  in  an  arc,  the  versed  sine  of  which  is 
equal  to  the  versed  sine  of  the  arc  described  by  the  point  of  at- 


MODE    OF    DRAWING   THE    PARALLEL    MOTION.         207 

tachment  on  the  side-rod.  As  the  bottom  of  the  side-rod  is  at- 
tached to  the  heam  and  the  top  to  the  cross-head,  and  as  the 
bottom  moves  in  an  arc  and  the  top  in  a  straight  line,  it  is  clear 
that  every  intermediate  point  of  the  side-rod  mnst  describe  an 
arc  which  -will  more  and  more  approach  to  a  straight  line,  or 
have  a  smaller  and  smaller  versed  sine,  the  nearer  such  point  is 
to  the  top  of  the  rod.  By  drawing  down  the  side-rod  at  the  end 
of  the  stroke,  and  also  at  half  stroke,  the  amount  of  deviation 
from  the  vertical  at  those  positions,  can  easily  be  determined  for 
any  point  in  the  length  of  the  rod ;  and  the  point  of  attachment 
of  the  parallel  bar  has  only  to  be  such,  and  the  length  and  travel 
of  the  radius  crank  has  also  to  be  such,  that  the  end  of  the  paral- 
lel bar  attached  to  the  side-rod  shall  describe  an  arc  whose  versed 
sine  is  equal  to  the  deviation  from  the  perpendicular,  or,  in  other 
words,  to  the  side-travel  of  that  point  of  the  side-rod  at  which 
the  attachment  is  made.  Since,  then,  the  side-rod  is  guided  at 
the  bottom  by  the  arc  of  the  beam,  and  near  the  top  by  that  less 
arc  described  by  the  end  of  the  parallel  bar,  which  answers  to 
the  supposition  of  the  cross-head  moving  in  a  vertical  line,  the 
result  is  that  the  cross-head  will  be  constrained  to  move  in  this 
vertical  line ;  since  only  on  that  supposition  can  the  two  arcs 
already  fixed  be  described. 

The  method  of  balancing  the  momentum  of  the  moving  parts 
of  marine  engines  which  I  introduced  in  1852  has  now  been 
very  generally  adopted ;  and  the  practice  is  found  to  be  very 
useful  in  reducing  the  tremor  and  uneasy  movements  to  which 
engines  working  at  a  high  rate  of  speed  are  otherwise  subject. 
Nearly  all  the  engines  now  employed  for  driving  the  screw  pro- 
peller are  direct-acting  engines,  which  necessarily  work  at  a  high 
rate  of  speed  to  give  the  requisite  velocity  of  rotation  to  the 
screw  shaft.  The  principle  on  which  the  balancing  is  effected  is 
that  of  applying  a  weight  to  the  crank  or  shaft,  and  when  the 
piston  and  its  connexions  move  in  one  direction  the  weight 
moves  in  the  opposite  with  an  equal  momentum. 


CHAPTER  IV. 

PROPORTIONS  OF  STEAM-ENGINES. 

WE  now  come  to  the  question  how  we  are  to  determine  the 
proportions  of  steam-engines  of  every  class. 

The  nominal  power  of  a  low  pressure  engine  is  determined 
by  the  diameter  of  the  cylinder  and  length  of  the  stroke,  as 
follows : — 


TO  DETERMINE  THE  NOMINAL  POWEE  OF  A  LOW  PEESStJKE  ENGINE 
OF  WATT'S  CONSTRUCTION. 

RULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  "by  the  cube  root  of  the  stroke  in  feet,  and  divide  the 
product  ~by  47.  The  quotient  is  the  nominal  horse-power  of 
the  engine. 

Example  1. — What  is  the  nominal  power  of  a  low  pressure 
engine  with  a  cylinder  64  inches  diameter  and  8-feet  stroke  ? 

Here  64  x  64  =  4,096,  which  multiplied  hy  2,  the  cube  root 
of  8  =  8,192  and  -*-  47  =  174-3. 

The  nominal  powers  of  engines  of  different  sizes,  both 
high  pressure  and  low  pressure,  are  given  in  the  following 
tables : — 


TABLES   OF   NOMINAL   POWERS   OF   ENGINE. 


209 


NOMINAL   HORSE   POWER   OF   HIGH   PRESSURE   ENGINES. 


^  >-  -: 

2?J 

1=1 
Sff 

0° 

LENGTH    OF   STEOKE   IN   FKET. 

1 

1| 

2 

2i 

3 

3i 

4 

5 

6 

7 

8 

9 

2 

•25 

•29 

•32 

•35 

•37 

•38 

•40 

•44 

•46 

•49 

•51 

•53 

2* 

•39 

•45 

•50 

•54 

•57 

•60 

•63 

•68 

•72 

•76 

•79 

•83 

8 

•57 

•65 

•72 

•78 

83 

•87 

•91 

•98 

1-04 

1-10 

115 

1-20 

8* 

•78 

•89 

•98 

1-06 

1-13 

1-19 

1-24 

1-34 

1-42 

1-49 

1-56 

1-G2 

4 

1-02 

1-17 

1-29 

1-38 

1-47 

1-56 

1-62 

1-74 

1-86 

1-95 

2-04 

2-10 

4* 

1-29 

1-48 

1-63 

1-75 

1-86 

1-96 

2-05 

2-21 

235 

2-47 

258 

2-68 

5 

1-59 

1-83 

2-01 

2-16 

2"28 

2-43 

2-52 

2-73 

2-88 

3-06 

818 

3-33 

5* 

1-93 

2-21 

2-43 

2-62 

2-78 

2-93 

3-12 

3-30 

8-51 

369 

3-86 

4-01 

6 

2-28 

2-01 

2-88 

8-12 

3-30 

3-48 

8-66 

3-93 

4-17 

4-41 

4-59 

4-77 

6* 

2-ey 

3-09 

3-39 

8-66 

8-90 

4-08 

4-23 

4-62 

4-89 

5-16 

5-46 

5-61 

7 

3-12 

8-57 

3-93 

4-23 

4-50 

4-74 

4-95 

5-34 

5-67 

5-97 

6"27 

6-51 

T* 

8-60 

4-11 

4-53 

4-86 

6-19 

5-46 

5-70 

6-15 

6-51 

6-87 

7-18 

7-46 

8 

4-08 

4-68 

5-16 

5-55 

5-88 

6-21 

(J-4-- 

6-99 

741 

7-80 

8-16 

8-49 

8* 

4-62 

5-28 

5-82 

6-27 

6-63 

6-99 

7-32 

7-9 

8-37 

8-82 

9-29 

9-59 

9 

5-16 

5-91 

6-51 

7-02 

7-47 

7-86 

8-22 

8-85 

9-39 

9-90 

10-35 

1077 

9* 

5-76 

6-60 

7-26 

7-SO 

8-37 

8-76 

9-15 

9-84 

10-47 

11-01 

11-52 

1-1-98 

10 

6-89 

7-32 

8-04 

8-67 

9-21 

9-69 

10-14 

10-92 

11-61 

12-21 

12-78 

13-29 

1% 

7-05 

8-04 

8-88 

9-54 

10-14 

10-68 

11-16 

12-03 

12-78 

13-47 

14-07 

14-64 

11 

7-74 

V-:, 

9-72 

10-47 

1181 

1173 

12-45 

13-20 

14-04 

14-76 

15-45 

1605 

11* 

8-43 

9-66 

10-62 

11-46 

12-15 

12-78 

18-80 

14-61 

15-88 

16-14 

16-88 

17-56 

12 

9-18 

10-53 

11-58 

12-48 

13-26 

13-95 

14-58 

15-72 

16-71 

17-58 

18-89 

1911 

12* 

9-96 

11-40 

12-57 

13-53 

14-87 

15-15 

15-84 

17-04 

18-12 

19-08 

1992 

20-73 

1ST 

10-80 

12-36 

13-59 

14-64 

15-57 

16-38 

16-92 

18-45 

19-59 

20-64 

21-57 

22-44 

18* 

11-64 

18-32 

14-64 

15-78 

16-77 

17-67 

18-48 

19-89 

21-15 

22-26 

2325 

2419 

1<T 

12-51 

14-81 

15-75 

16-98 

18-03 

18-99 

19-86 

•J  l-8'.l 

22-74 

23-94 

25-02 

26-01 

14* 

18-41 

15-66 

16-92 

18-21 

19-35 

20-37 

21-80 

22-95 

24-89 

25-62 

26-83 

27-90 

15 

14-31 

16-44 

18-09 

19-50 

20-70 

21-81 

22-80 

24-57 

26-10 

27-48 

28-71 

29-88 

16 

16-35 

18-69 

20-58 

22-17 

23-58 

24-81 

25-95 

27-93 

29-70 

81-26 

82-67 

33-99 

17 

18-45 

21-12 

23-25 

25-05 

26-58 

28-02 

29-28 

81-56 

83-57 

35-28 

36-90 

8837 

18 

20-67 

28-67 

26-04 

28-08 

29-82 

81-41 

82-82 

85-37 

87-59 

89-57 

41-37 

48-02 

19 

23-04 

26-37 

29-04 

81-26 

88-51 

84-98 

86-57 

89-39 

41-88 

44-07 

46-08 

47-94 

20 

25-53 

29-22 

32-16 

84-65 

86-81 

8876 

40-53 

43-65 

46-38 

48-84 

51-06 

53-10 

22 

80-90 

85-37 

88-91 

41-94 

44-55 

46-89 

49-86 

52-95 

66-13 

59-10 

61-80 

6426 

24 

86-78 

42-09 

46-32 

49-89 

63-01 

55-88 

68-35 

62-85 

66-81 

70-82 

7353 

76-47 

26 

48-17 

49-38 

64-36 

58-56 

62-25 

65-52 

67-68 

73-80 

7842 

82-53 

86-84 

89-76 

28 

50-04 

57-27 

63-06 

67-92 

72-18 

75-99 

79-44 

85-56 

90-98 

95:70 

100-1 

104-0 

80 

57-45 

65-76 

72-89 

77-97 

82-86 

87-21 

91-20 

98-22 

104-4 

109-9 

114-9 

1195 

82 

65-87 

74-88 

82-53 

88-71 

94-26 

99-24 

108-7 

111-8 

118-7 

125-0 

180-7  1136-0 

84 

78-80 

84-48 

92-97 

100-2 

106-8 

112-0 

117-1 

126-2 

184-0 

141-1 

147-6 

158-5 

86 

82-71 

94-68 

104-2 

112-2 

119-8 

1266 

181-8 

141-4 

150-8 

158-2 

165-4 

172-1 

83 

92-16 

105-5 

116-1 

125-0 

134-0 

186-9 

146-3 

157-6 

167-5 

176-8 

184-8   191-7 

40 

102-1 

116-9 

129-6 

188-6 

147-8 

155-1 

162-1 

174-6 

185-6 

195-8 

204-2   2124 

42 

112-6 

128-9 

141-8 

152-8 

162-4 

170-9 

178-7 

192-5 

204-6 

215-8 

225-2   284-2 

44 

123-6 

141-4 

155-7 

167-7 

178-1 

1S7-6 

199-4 

211-8 

224-5 

236-8 

247-1 

257-0 

46 

185-0 

154-6 

170-1 

183-3 

194-6 

204-6 

214-8 

280-0 

245-4 

258-8 

2701 

fto-g 

48 

147-0 

168-8 

185-3 

199-6 

212-1 

2232 

283-4 

251-8 

267-2 

281-8 

294-1   306-0 

60 

159-6 

182-6 

201-0 

216-5 

280-1 

242-8 

253-3 

272-9 

289-9 

805-1 

819-2   331-8 

62 

172-6 

197-6 

217-4 

234-2 

249-0 

262-0 

270-7 

295-2 

818-5 

380-0 

845-8   8588 

64 

186-1 

218-0 

234-5 

252-6 

268-4 

282-6 

295-4 

8188 

888-1 

856-1 

372-3   3870 

56 

2CO-1 

229-1 

252-2 

271-6 

288-7 

303-9 

8177 

842-3 

368-6 

••':•*•>•* 

400-2   416-4 

68 

214-7 

245-8 

270-5 

291-4 

309-6 

825-8 

340-8 

867-2 

8897 

410-1 

429-8   446-1 

60 

229-8 

263-0 

289-5 

811-7 

881-2 

848-9 

364-8 

898-0 

417-6 

4896 

4596   477-9 

70 

312-8 

857-9 

393-9 

424-5 

451-2 

474-9 

496-5 

5846 

6682 

598-2 

625-5   650-4 

210 


PROPORTIONS    OF    STEAM-ENGINES. 


NOMINAL    HORSE    POWER   OF   LOW    PRESSURE    ENGINES. 


*•'*    00 

"-=•§ 
EO 

LENGTH   OF   STROKE    IN    FEET. 

1 

H 

2 

01 
*2 

3 

3* 

4 

5 

6 

7 

8 

9 

4 

•35 

•39 

•43 

•46 

•49 

•52 

•51 

•58 

•62 

•65 

•G8 

•70 

5 

•53 

•61 

•67 

•72 

•76 

•81 

.84 

•91 

•96 

1-02 

1-06 

1-14 

6 

•76 

•87 

•96 

1-04 

1-10 

1-16 

1-22 

1-31 

1-39 

1-47 

1-53 

1*59 

7 

1-04 

1-19 

1-31 

1-41 

1-50 

1-58 

1-65 

1-78 

1-89 

1-99 

2-09 

2-17 

8 

1-36 

1-56 

1-72 

1-85 

1-96 

2.07 

2-16 

2-33 

2-47 

2-60 

2-72 

2-83 

9 

1-72 

1-97 

2-17 

2-34 

2-49 

2-62 

2-74 

2-95 

3-13 

3-30 

8-45 

3-59 

10 

2-13 

2-44 

2-68 

2-S9 

3-07 

3-23 

3-38 

8-64 

3-87 

4-07 

4-26 

4-48 

11 

2-57 

2-95 

8-24 

3-49 

3-77 

3-91 

4-15 

4-40 

4-68 

4-92 

5-15 

5-35 

12 

3-06 

3-51 

8-86 

4-16 

4-42 

4-65 

4-66 

5-24 

5-57 

5-86 

6-18 

6-37 

13 

3-60 

4-12 

4-53 

4-68 

5-19 

5-46 

5-64 

6-15 

6-53 

6-88 

7-19 

7-48 

14 

4-17 

4-77 

5-25 

5-66 

6-01 

6-33 

6-02 

7-13 

7-58 

7-98 

8-34 

8-67 

15 

4-77 

5-48 

6-03 

6-50 

6-90 

7-27 

7-60 

8-19 

8-70 

9-16 

9-57 

9-96 

16 

5-45 

6-23 

6-86 

7-39 

7-86 

8-27 

8-65 

9-31 

9-90 

10-42 

10-89 

11-33 

IT 

6-15 

7-04 

7-75 

8-35 

8-86 

9-34 

9-76 

10-52 

11-17 

11-76 

12-30 

12-79 

18 

6-S9 

7-89 

8-68 

9-36 

9-94 

10-47 

10-94 

11-79 

12-53 

13-19 

13-79 

14-34 

19 

7'68 

8-79 

9-68 

10.42 

11-17 

11-66 

12-19 

13-13 

13-96 

14-69 

15-36 

15-98 

20 

8-51 

9-74 

10-72 

11-55 

12-27 

12-92 

13-51 

14-55 

15-46 

16-28 

17-02 

17-70 

22 

10-30 

11-79 

12-97 

13-98 

14-85 

15-63 

16-62 

17-65 

18-71 

19-70 

20-60 

21-42 

24 

12-26 

14-03 

15-44 

16-63 

17-67 

18-61 

19-45 

20-95 

22-27 

23-44 

24-51 

25-49 

26 

14-39 

16-46 

18-12 

19-52 

20-75 

21-84 

22-56 

24-60 

26-14 

27-51 

28-78 

29-92 

28 

16-68 

19-09 

21-02 

22-64 

24-06 

25-33 

26-48 

28-52 

80-31 

81-90 

33-36 

84-69 

80 

19-15 

21-92 

24-13 

25-99 

27-62 

20-07 

80-40 

32-74 

34-80 

36-63 

38-30 

89-83 

82 

21-79 

24-96 

27-51 

29-57 

31-42 

83-08 

34-59 

87-26 

39-59 

41-6& 

43-57 

45-32 

84 

24-60 

28-16 

30-99 

33-39 

35-44 

37-34 

39-04 

42-06 

44-69 

47-05 

49-19 

61-16 

86 

27-57 

31-56 

34-74 

87-42 

89-77 

41-87 

43-77 

47-15 

50-11 

52-75 

65-15 

57-36 

88 

30-72 

35-17 

38-71 

41-69 

44-66 

46-64 

48-77 

52-54 

55-83 

58-78 

61-45 

63-91 

40 

84-04 

88-97 

42-89 

46-20 

49-10 

51-69 

54-04 

58-21 

61-86 

65-12 

68-08 

70-81 

42 

37-53 

42-96 

47-29 

50-94 

54-13 

56-98 

59-58 

64-18 

68-21 

71-78 

75-06 

78-06 

44 

41-19 

47-15 

51-90 

55-91 

59-38 

62-54 

66-46 

70-44 

74-85 

78-79 

82-38 

85-68 

46 

45-02 

51-54 

56-72 

61-10 

64-88 

68-19 

71-48 

76-69 

81-81 

86-12 

90-04 

93-64 

48 

49-02 

56-11 

61-76 

66-53 

70-70 

74-42 

77-82 

83-83 

89-08 

93-78 

98-04 

102-0 

50 

53-19 

60-89 

67-02 

72-19 

76-71 

80-76 

84-44 

90-96 

96-65 

101-7 

106-4 

110-6 

52 

57-55 

65-86 

72-48 

78-08 

83-00 

87-35 

90-25 

98-40 

104-5 

110-0 

115-1 

119-6 

54 

62-04 

71-02 

78-17 

84-20 

89-48 

94-20 

98-49 

106-1 

112-7 

118-7 

124-1 

129-0 

56 

66-72 

76-88 

84-07 

90-55 

96-23 

101-30 

105-9 

114-1 

121-2 

127-6 

133-4 

138-8 

58 

71-58 

81-93 

9018 

97-14 

103.2 

108-6 

113-6 

122-4 

129-9 

136-7 

143-1 

148-7 

60 

76-60 

87-68 

96.50 

108-9 

110-4 

116-8 

121-6 

181-0 

139-2 

146-5 

153-2 

159-8 

62 

81-79 

93-62 

103-04 

111-0 

117.9 

124-18 

129-81 

139-8 

148-6 

156-7 

163-6 

170-8 

64 

87-15 

99-84 

110-0 

118-8 

125-7 

182-3 

138-8 

149-0 

158-4 

166-7 

174-8 

181-3 

66 

92-68 

106-1 

116-8 

125-8 

133-6 

140-7 

147-3 

158-5 

168-4 

177-8 

185-4 

192-8 

68 

98-40 

112-6 

123-9 

133-6 

141-8 

149-4 

1562 

168-2 

178-8 

188-2 

196-8 

204-6 

70 

104-26 

119-3 

181-3 

141-5 

150-4 

158-8 

165-5 

178-2 

189-4 

199-4  i  208-5 

216-8 

72 

110-30 

126-2 

139-0 

149-7 

159-1 

167-4 

175-1 

188-6 

200-4 

211-0    220-6 

229-4 

74 

116-5 

133-4 

146-8 

158-1 

167-9 

176-7 

185-4 

199-2 

211-6 

223-4  ;  288-0 

242-2 

76 

122-9 

140-7 

154-8 

166-8 

178-6 

186-6 

195-0 

210-1 

223-3 

285-1    245-8 

255-6 

7S 

129-4 

148-2 

163-1 

175-6 

186-7 

196-5 

205-4 

221-4 

235-2 

247-6    258-9  1269-2 

80 

136-2 

155-8 

171-6 

184-8 

196-4 

206-7 

216-1 

282-8    247-4 

260-5  i  272-8    283'2 

82 

143-0 

163-8 

180-2 

194-2 

206-2 

217-8 

226-9 

244-6    260-0 

278-8   286-1  ;297'6 

84 

150-1 

171-8 

189-1 

208-8 

216-5 

227-9 

288-3    256-7    272-8 

287-1    800-2   812-2 

86 

157-4 

180-1 

198-2 

218-6 

227-0 

237-8 

247-4   269-1    286-0  '801-0   314-7    327'3 

88 

164-8 

188-6 

207-6 

228-6  1287-5  1  250-2 

261-6 

281-7    299-4   815-2   829-5   842-7 

90 

172-3 

197-3 

2171 

238-9 

248-6 

261-7 

273-6 

294-7    813-2   329-7    344-7    858-5 

100    212-8 

243-5 

268-0 

288-8 

806.8 

828-0 

887-7 

863-8    886-6   407-0   425-5   442'6 

RULES    FOR   FINDING   THE   HORSES    POWER.  211 

Example  2. — What  is  the  nominal  power  of  a  low  pressure 
engine  of  40  inches  diameter  of  cylinder  and  5-feet  stroke. 

Here  40  x  40  =  1,600,  which  multiplied  by  1'71 — which  is 
the  cube  root  of  5  very  nearly — we  get  2,736,  which  divided  by 
47  gives  5S-21  as  the  nominal  horse  power. 

The  actual  horse  power  of  an  engine  is  determinable  by  the 
application  of  an  instrument  to  determine  the  amount  of  power 
it  actually  exerts.  The  mode  of  determining  this  will  be  ex- 
plained hereafter.  Meanwhile  it  may  be  repeated  that  an  actual 
horse  power  is  a  dynamical  unit  capable  of  raising  a  load  of 
33,000  Ibs.  one  foot  high  in  each  minute  of  time.  The  nominal 
poicer  of  a  high  pressure  engine  maybe  taken  at  three  times 
that  of  a  low  pressure  engine  of  the  same  size. 

The  assumed  pressure  in  computing  the  nominal  power  of 
low  pressure  engines  is  7  Ibs.  on  each  square  inch  of  the  piston, 
and  the  assumed  pressure  in  computing  the  nominal  power  of 
high  pressure  engines  is  21  Ibs.  on  each  square  inch  of  the  piston. 
The  assumed  speed  of  the  piston  varies  with  the  length  of  stroke 
from  160  to  256  feet  per  minute,  namely,  for  a  2  ft.  stroke, 
160  ft. ;  23- ft.,  170 ;  3  ft.,  180 ;  4  ft,  200 ;  5  ft,  215 ;  6  ft,  228 ; 
7  ft.,  245 ;  and  8  ft.,  256  feet  per  minute. 

In  point  of  fact,  in  all  modern  low  pressure  engines  the  un- 
balanced pressure  of  steam  upon  the  piston  is  much  more  than 
7  Ibs.,  and  in  most  modern  high  pressure  engines  the  unbalanced 
pressure  of  steam  upon  the  piston  is  much  more  than  21  Ibs.  The 
speed  of  the  piston  is  also  frequently  much  more  than  256  feet 
per  minute.  In  the  case  of  screw  engines  the  Admiralty  employs 
a  rule  to  determine  the  power,  in  which  the  old  assumed  pressure 
of  7  Ibs.  per  square  inch  is  retained,  but  in  which  the  actual  speed 
of  piston  is  taken  into  account.  This  rule  is  as  follows : — 

ADMIBALTY   RULE  FOB  DETERMINING   THE   NOMINAL   POWEE   OF   AN 
ENGINE. 

RULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  by  the  speed  of  the  piston  in  feet  per  minute,  and 
divide  by  6.000.     The  quotient  is  the  nominal  power. 
Example. — What  ia  the  power  of  an  engine  with  a  cylinder 


212  PROPORTIONS   OF   STEAM-ENGINES. 

of  42  inches  diameter,  and  3-J  feet  stroke,  and  which  makes  85 
revolutions  per  minute  ? 

Here  42  x  42  —  1,764.  The  length  of  a  double  stroke  will 
be  3-J-  x  2  —  7  feet,  and  as  there  are  85  revolutions  or  double 
strokes  per  minute,  85  x  7  =  595  will  be  the  speed  of  the  piston 
in  feet  per  minute.  Now  1,764  x  595  =  1,049,580,  which,  di- 
vided by  6,000  =  175  horses  power. 

The  area  of  the  piston  in  circular  inches,  it  will  be  recollect- 
ed, is  found  by  multiplying  the  diameter  by  itself.  Thus  a  pis- 
ton 50  inches  diameter  contains  50  x  50,  or  2,500  circular  inches. 
Now  as  every  circular  inch  is  '7854  of  a  square  inch,  we  must, 
in  order  to  find  the  area  of  the  piston  in  square  inches,  multiply 
the  diameter  by  itself  and  by  '7854,  which  will  give  the  area  in 
square  inches.  Thus,  2,500  x  '7854  =  1,963'5  square  inches, 
which  is  the  area  in  square  inches  of  a  piston  50  inches  in  diam- 
eter. The  circumference  of  any  circle  is  obtained  by  multiply- 
ing the  diameter  by  31416.  Hence  the  length  of  a  string  or 
tape  that  will  be  required  to  encircle  a  piston  50  inches  in  diam- 
eter will  be  50  x  3'1416  =  157'08  inches.  The  areas  of  pumps, 
pipes,  safety-valves,  and  all  other  circular  objects,  is  computed  in 
the  same  way  as  the  areas  of  circles  or  pistons.  Some  valves 
are  annular  valves,  consisting  not  of  a  flat  circular  plate,  but  of 
a  ring  or  annulus  of  a  certain  breadth.  To  compute  the  area 
of  such  a  valve  we  must  first  compute  the  area  of  the  outer 
circle,  and  then  the  area  of  the  inner,  and  subtract  the  less  from 
the  greater,  which  will  give  the  area  of  the  annulus.  So  in  like 
manner,  in  trunk  engines,  we  must  subtract  the  area  of  the 
trunk  from  the  area  of  the  piston. 

GENEKAL  CONSIDERATIONS  AND  INSTRUCTIONS. 

In  proceeding  to  design  an  engine  for  any  given  purpose, 
the  nominal  power  may  either  be  fixed  or  the  nominal  power 
may  be  left  indeterminate,  and  only  the  work  be  fixed  which 
the  engine  has  to  perform.  In  the  first  case  we  have  only  to 
ascertain  by  the  foregoing  rules  or  tables  what  the  dimensions 
of  a  cylinder  are  which  correspond  to  the  nominal  power, 


DRAWINGS    SUITABLE   FOR   AXL   POWERS.  213 

and  we  have  then  to  make  all  the  other  parts  of  dimensions 
corresponding  thereto,  •which  we  shall  be  enabled  to  do  by  the 
rules  here  laid  down.  Of  course  the  engineer  settles  for  himself 
some  particular  type  of  engine  which  he  prefers  to  adopi  as  the 
one  that  is  to  govern  his  practice,  and  any  drawing  of  an  engine 
of  a  given  size  or  power  is  applicable  to  the  construction  of  a 
similar  engine  of  any  other  size  or  power  by  merely  altering  the 
scale  of  the  drawing.  If,  therefore,  any  engineer  decides  upon 
the  class  of  land  engine,  paddle  engine,  or  screw  engine  which 
he  prefers  to  construct,  and  chooses  to  get  a  set  of  drawings  of 
such  engine  on  any  given  scale  lithographed,  such  drawir^s  will 
be  applicable  to  all  sizes  and  powers  of  that  class  of  engine  by 
altering  the  scale  in  the  proportion  rendered  necessary  by  the 
enlarged  or  diminished  diameter  of  the  cylinder  answerable  to 
the  required  power.  Thus,  if  we  have  a  drawing  of  a  marine 
engine  of  32  inches  diameter  of  cylinder  and  4-feet  stroke,  made 
to  the  scale  of  -J-inch  to  the  foot,  we  may  from  such  drawing 
construct  a  similar  engine  of  64  inches  diameter  and  8-feet 
stroke  by  merely  altering  the  scale  to  one  of  ^-inch  to  the  foot, 
so  that  every  part  will  in  fact  measure  twice  what  it  measured 
before.  In  order  to  make  the  same  drawing  applicable  to  any 
size  of  engine,  whether  large  or  small,  we  have  only  to  divide 
the  diameter  of  the  cylinder  into  the  number  of  parts  that  the 
cylinder  is  to  have  of  inches,  and  then  we  may  use  the  scale  so 
formed  for  the  scale  of  the  drawing.  Thus,  if  we  wish  the 
engine  to  have  a  cylinder  of  30  inches  diameter,  we  must  divide 
the  diameter  of  the  cylinder  as  shown  in  the  drawing  into  30 
equal  parts,  each  of  which  will  represent  an  inch,  and  of  course  any 
twelve  of  them  will  represent  a  foot.  If  we  now  measure  any 
other  part  of  the  engine,  such  as  the  diameter  of  the  air  pump, 
diameter  of  crank  shaft,  or  any  other  part  by  this  scale,  we  shall 
find  the  proper  dimensions  of  the  part  in  question,  If  we  wish 
to  construct  from  the  drawing  an  engine  of  60  or  100  inches, 
and  of  corresponding  stroke,  we  have  only  to  divide  the  diam- 
eter of  the  cylinder  into  60  or  100  equal  parts,  and  use  each  of 
those  parts  as  an  inch  of  the  scale,  when  the  proper  dimensions 
of  all  the  parts  will  be  at  once  obtained. 


214  PROPORTIONS    OF    STEAM-ENGINES. 

It  will  be  needless  to  guard  these  remarks  against  the  obvious 
exception  that  in  case  of  very  large  and  very  small  engines  it 
will  be  proper  to  make  such  slight  modifications  in  some  of  the 
details  as  will  conduce  to  greater  convenience  in  working  or  in 
construction.  For  instance,  as  the  height  and  strength  of  a  man 
are  a  given  quantity,  it  will  obviously  not  be  proper  in  doubling 
the  size  of  all  the  other  parts  to  double  the  height  of  the  starting 
handles,  or  even  to  double  their  strength.  In  the  case  of  oscil- 
lating engines,  again,  with  a  crank  in  the  intermediate  shaft,  it 
may  be  difficult  to  get  a  sound  crank  made  in  the  case  of  very 
large  engines,  and  some  other  expedient  may  have  to  be  adopted. 
Again,  in  the  case  of  very  small  engines,  the  flanges  and  bolts 
may  require  to  be  a  little  larger  than  the  proportion  derived 
from  a  drawing  of  large  engines,  and  the  valve  chests  of  the 
feed  pumps  and  other  parts  may  be  too  small  if  made  strictly  to 
scale  to  get  the  hand  into  conveniently  to  clear  them  out.  All 
such  points  however  are  matters  of  practical  convenience,  only 
to  be  determined  by  the  thoughtfulness  and  experience  of  the 
engineer,  and  in  nowise  affect  the  main  conclusion  that  a  draw- 
ing of  an  engine  of  any  one  size  will  suffice  for  the  construction 
of  engines  of  other  sizes  by  merely  changing  the  scale.  It  will 
consequently  save  much  trouble  in  drawing  offices  to  have  one 
certain  type  of  engine  of  each  kind  lithographed  in  all  its  details, 
and  then  engines  of  all  sizes  may  be  made  therefrom  by  adding 
the  proper  scale,  and  by  marking  upon  the  drawing  the  proper 
dimensions  of  each  part  in  feet  or  inches — the  measurements 
being  taken  from  a  table  fixed  once  for  all,  either  by  computation 
or  by  careful  measurement  of  the  drawing  with  the  different 
suitable  scales.  By  thus  systematising  the  work  of  the  drawing 
office,  labour  may  be  saved  and  mistakes  prevented. 

It  easy  to  understand  the  principle  on  which  the  main  parts 
of  an  engine  must  be  proportioned.  "We  must  in  the  first  place 
have  the  requisite  quantity  of  boiler  surface  to  generate  the 
steam,  the  requisite  quantity  of  water  sent  into  the  boiler  to 
keep  up  the  proper  supply,  and  the  requisite  quantity  of  cold 
water  to  condense  the  steam  after  it  has  given  motion  to  the 
piston.  In  common  boilers  about  10  square  feet  of  heating 


GENERAL    CONSIDERATIONS    AND    INSTRUCTIONS.       215 

surface  will  boil  off  a  cubic  foot  of  water  in  the  hour,  and  this 
in  the  older  class  of  engines  was  considered  the  equivalent  of  a 
horse  power.  At  the  atmospheric  pressure,  or  with  no  load  on 
the  safety  valve,  a  cubic  inch  of  water  makes  about  a  cubic  foot 
of  steam ;  and  at  twice  the  atmospheric  pressure,  or  with  15  Ibs. 
per  square  inch  on  the  safety  valve,  a  cubic  inch  of  water  will 
make  about  half  a  cubic  foot  of  steam.  For  every  half  cubic 
foot  of  such  steam  therefore  abstracted  from  the  boiler  there 
must  be  a  cubic  inch  of  water  forced  into  it.  So  if  we  take  the 
latent  heat  of  steam  in  round  numbers  at  1,000  degrees,  and  if 
the  condensing  water  enters  at  60°,  and  escapes  at  100°,  the 
condensing  water  has  obviously  received  40  degrees  of  heat,  and 
it  has  received  this  from  the  steam  having  1,000°  of  heat,  and 
the  112°  which  the  steam  if  condensed  into  boiling  water  would 
exceed  the  waste-water  in  temperature.  It  follows  that  in  order 
to  reduce  the  heat  of  the  steam  to  100°  there  must  be  1,112°  of 
heat  extracted,  and  if  the  condensing  water  was  only  to  be 
heated  1  degree,  there  would  require  to  be  1,112  tunes  the 
quantity  of  condensing  water  that  there  is  water  in  the  steam. 
Since,  however,  the  water  is  to  be  heated  40°,  there  will  only 
require  to  be  one-fortieth  of  this,  or  about  ^th  the  quantity  of 
injection  water  that  there  is  water  in  the  steam.  These  rough 
determinations  will  enable  the  principle  to  be  understood  on 
which  such  proportions  are  determined.  The  proportions  of  the 
condenser  and  of  the  air-pump  were  determined  by  Mr.  Watt 
at  one- eighth  of  the  capacity  of  the  cylinder.  In  more  modern 
engines,  and  especially  in  marine  engines  where  there  are  irregu- 
larities of  motion,  the  air-pump  is  generally  made  a  little  larger 
than  this  proportion,  and  with  advantage.  The  condenser  is 
also  generally  made  larger,  and  many  engineers  appear  to  con- 
sider that  the  larger  the  condenser  is  the  better.  Mr.  Watt, 
however,  found  that  when  the  condenser  was  made  larger  than 
one-eighth  of  the  capacity  of  the  engine  the  efficiency  of  the 
engine  was  diminished.  The  fly-wheel  employed  in  land  engines 
to  control  the  irregularities  of  motion  that  would  otherwise 
exist,  is  constructed  on  the  principle  that  there  shall  be  a  revolv- 
ing mass  of  such  weight,  and  moving  with  such  a  velocity,  as  to 


216  PROPORTIONS   OP   STEAM-ENGINES. 

constitute  an  adequate  reservoir  of  power  to  redress  irregulari- 
ties. It  is  found  that  in  those  cases  where  the  most  equable 
motion  is  required,  it  is  proper  to  have  as  much  power  treasured 
up  in  the  fly-wheel  as  is  generated  in  6  half-strokes,  though  in 
many  cases  the  proportion  is  not  more  than  half  this.  It  is  quite 
easy  to  tell  what  the  weight  and  velocity  of  the  fly-wheel  must 
be  to  possess  this  power.  When  we  know  the  area  of  the  piston 
and  the  unbalanced  pressure  per  sq.  inch,  we  easily  find  the 
pressure  urging  it,  and  this  pressure  multiplied  by  the  length  of 
6  half-strokes  represents  the  amount  of  power  which,  in  the 
most  equable  engines,  the  fly-wheel  must  possess.  Thus,  suppose 
that  the  pressure  on  the  piston  were  a  ton,  and  that  the  length 
of  the  cylinder  were  5  feet,  then  in  6  half-strokes  the  space 
described  by  the  piston  would  be  30  feet.  The  measure  of  the 
power  therefore  is  1  ton  descending  through  30  feet,  and  if  there 
were  any  circumstance  which  limited  the  weight  of  the  fly- 
wheel to  1  ton,  then  the  velocity  of  the  rim — or  more  correctly 
of  the  centre  of  gyration — must  be  equal  to  that  which  any 
heavy  body  would  have  at  the  end  of  the  descent  by  falling  from 
a  height  of  30  feet,  and  which  velocity  may  easily  be  determined 
by  the  rule  already  given  for  ascertaining  the  velocity  of  falling 
bodies.  If  the  weight  of  the  fly-wheel  can  be  2  tons,  then  the 
velocity  of  the  rim  need  only  be  equal  to  that  of  a  body  falling 
through  15  feet,  and  so  in  all  other  proportions,  so  that  the 
weight  and  velocity  can  easily  be  so  adjusted  as  to  represent 
most  conveniently  the  prescribed  store  of  power. 

With  these  preliminary  remarks  it  will  now  be  proper  to 
proceed  to  recapitulate  the  rules  for  proportioning  all  the  parts 
of  steam  engines  illustrated  by  examples : — 

STEAM  PORTS. 

The  area  of  steam  port  commonly  given  in  the  best  engines 
working  at  a  moderate  speed  is  about  1  square  inch  per  nominal 
horse-power,  or  ^yth  of  the  area  of  the  cylinder,  and  the  area  of 
the  steam  pipe  leading  into  the  cylinder  is  less  than  this,  or  -66 
square  inch  per  nominal  horse  power.  Since  however  engines 


PROPER   AREAS    OP   CYLINDER   PORTS.  217 

are  now  worked  at  various  rates  of  speed  it  will  be  proper  to 
adopt  a  rule  in  which  the  speed  of  the  piston  is  made  an  element 
of  the  computation.  This  is  done  in  the  rules  which  follow  both 
for  the  steam  port  and  branch  steam  pipe. 

TO  FIND  THE  PEOPEB  AREA  OF  THE  STEAM  OE  EDUCTION  POET 
OF  THE  CYLIXDEE. 

EULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  l>y  the  speed  of  the  piston  in  feet  per  minute  and  l>y 
the  decimal  '032,  and  divide  the  product  ly  140.  The  quo- 
tient is  the  proper  area  of  the  cylinder  port  in  square  inches. 

Example.— ~Wh&t  is  the  proper  area  of  each  cylinder  port  in 
an  engine  with  64-inch  cylinder,  and  with  the  piston  travelling 
220  feet  per  minute  ? 

Here  64  x  64  =  4,096,  which  multiplied  by  220  =  901,120, 
and  this  multiplied  by  '032  =  28,835-8,  which  divided  by  140,  gives 
206  inches  as  the  area  of  each  cylinder  port  in  square  inches. 

This  is  a  somewhat  larger  proportion  than  is  given  in  some 
excellent  engines  in  practice.  But  inasmuch  as  the  application 
of  lap  to  the  valve  virtually  contracts  the  area  of  the  cylinder 
ports,  and  as  the  application  of  such  lap  is  now  a  common  prac- 
tice, it  is  desirable  that  the  area  of  the  ports  should  be  on  the 
large  side.  In  the  engines  of  the  'Clyde,'  'Tweed,'  'Tay,'  and 
'  Teviot,'  by  Messrs.  Oaird  and  Co.,  the  diameter  of  the  cylinder 
was  74|  inches,  and  the  length  of  the  stroke  TJ  feet,  so  that  the 
nominal  power  of  each  engine  was  about  234  horses.  The  cyl- 
inder ports  were  33J-  inches  long  and  6|  inches  broad,  so  that 
the  area  of  each  port  was  224 '4  square  inches,  being  somewhat 
less  than  the  proportion  of  1  square  inch  per  nominal  horse 
power,  but  somewhat  more  than  the  proportion  of  ^th  of  the 
area  of  the  cylinder.  As  the  areas  of  circles  are  in  the  propor- 
tion of  the  square  roots  of  their  respective  diameters,  the  area 
of  a  circle  of  one-fifth  of  the  diameter  of  the  piston  will  have 
one-twenty-fifth  of  the  area  of  the  piston.  One-fifth  of  74f  ths 
is  15  nearly,  and  the  area  of  a  circle  15  inches  in  diameter  is 
176'7  square  inches,  which  is  considerably  less  than  the  actual 
10 


218  PROPORTIONS   OF   STEAM-ENGINES. 

area  of  the  port.  By  the  rule  we  have  given  the  area  of  the 
ports  of  this  engine  would,  at  a  speed  of  220  feet  per  minute,  be 
about  277"  square  inches,  which  is  somewhat  greater  than  the 
actual  dimensions.  At  a  speed  of  the  piston  of  440  feet  per 
minute  the  area  of  the  port  would  be  double  the  foregoing. 

STEAM  PIPE. 

In  the  engines  already  referred  to,  the  internal  diameter  of 
each  steam  pipe  leading  to  the  cylinder  is  13-f-  inches,  which 
gives  an  area  of  145-8  square  inches.  It  is  not  desirable  to  make 
the  steam  pipe  larger  than  is  absolutely  necessary,  as  an  increased 
external  surface  causes  increased  loss  of  heat  from  radiation. 
The  following  rule  will  give  the  proper  area  of  the  steam  pipe 
for  all  speeds  of  piston: — 

TO   FIND   THE   AREA    OF   THE   STEAM  PIPE    LEADING   TO    EACH 
OTLINDEE. 

KULE. — Multiply  the  square  of  the  diameter  of  tlie  cylinder  in 
inches  by  the  speed  of  the  piston  in  feet  per  minute  and  ly  the 
decimal  '02,  and  divide  the  product  ~by  170.  The  quotient  is 
the  proper  area  of  the  steam  pipe  leading  to  the  cylinder  in 
inches. 

Example. — What  is  the  proper  area  of  the  branch  steam  pipe 
leading  to  each  cylinder  in  an  engine  with  a  cylinder  74J  inches 
diameter,  and  with  the  piston  moving  at  a  speed  of  220  feet  per 
minute? 

Here  74-6  x  74-5  =  5,550-25,  which  multiplied  by  220  = 
1,221,055.  and  this  multiplied  by  '02  =  24,421-1,  which  divided 
by  170  =  144  square  inches  nearly.  The  diameter  of  a  circle  of 
144  square  inches  area  is  a  little  over  13£  inches,  so  that  13£ 
inches  would  be  the  proper  internal  diameter  of  each  branch 
steam  pipe  in  such  an  engine.  The  main  steam  pipe  em- 
ployed in  steamers  usually  transmits  the  steam  for  both  the 
engines  to  the  end  of  the  engine-house,  where  it  divides  into 
two  branches — one  extending  to  each  cylinder.  The  main  steam 
pipe  will  require  to  have  nearly,  but  not  quite,  double  the 
area  of  each  of  the  branch  steam  pipes.  It  would  require  to 


PROPER  AREA  OF  SAFETY  VALVES.         219 

have  exactly  double  the  area,  only  that  the  friction  in  a  large 
pipe  is  relatively  less  than  in  a  small;  and  as,  moreover,  the 
engines  work  at  right  angles,  so  that  one  piston  is  at  the  end  of 
its  stroke  when  the  other  is  at  the  beginning,  and  therefore 
moving  slowly,  it  will  follow  that  when  one  engine  is  making 
the  greatest  demand  for  steam  the  other  is  making  very  little, 
so  that  the  area  of  the  main  steam  pipe  will  not  require  to  be  as 
large  as  if  the  two  engines  were  making  their  greatest  demand 
at  the  same  time. 

SAFETY  VALVES. 

It  is  easy  to  determine  what  the  size  of  an  orifice  should  be 
in  a  boiler  to  allow  any  volume  of  steam  to  escape  through  it  in 
a  given  time.  For  if  we  take  the  pressure  of  the  atmosphere  at 
15  Ibs.,  and  if  the  pressure  of  the  steam  in  the  boiler  be  10  Ibs. 
more  than  this,  then  the  velocity  with  which  the  steam  will  flow 
out  will  be  equal  to  that  which  a  heavy  body  would  acquire  in 
falling  from  the  top  of  a  column  of  the  denser  fluid  that  is  high 
enough  to  produce  the  greater  pressure  to  the  top  of  a  column 
of  the  same  fluid  high  enough  to  produce  the  less  pressure,  and 
this  velocity  can  easily  be  ascertained  by  a  reference  to  the  law 
of  falling  bodies.  In  practice,  however,  the  area  of  safety  valves 
is  made  larger  than  what  answers  to  this  theoretical  deduction, 
partly  in  consequence  of  the  liability  of  the  valves  to  stick  round 
the  rim,  and  because  the  rim  or  circumference  becomes  relatively 
less  in  the  case  of  large  valves.  One  approximate  rule  for  safety 
valves  is  to  allow  one  square  inch  of  area  for  each  inch  in  the  di- 
ameter of  the  cylinder,  so  that  an  engine  with  a  64-inch  cylinder 
would  require  a  safety  valve  on  the  boiler  of  64  square  inches 
area,  which  answers  to  a  diameter  of  about  9  inches.  The  rule 
should  also  have  reference,  however,  to  the  velocity  of  the  piston, 
and  this  condition  is  observed  in  the  following  rule: — 

TO   FIND   THE   PROPER  DIAMETER  OF  A  SAFETY  VALVE   THAT  WILL 
LET  OFF  ALL  THE  STEAM  FEOM  A  LOW  PRESSURE  BOILER. 

RULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
~by  the  speed  of  the  piston  in  feet  per  minute,  and 


220  PROPORTIONS    OF   STEAM-ENGINES. 

divide  the  product  ly  14,000.     The  quotient  is  the  proper 
area  of  the  safety  valve  in  square  indies. 

Example. — "What  is  the  proper  diameter  of  the  safety  valve 
of  a  boiler  that  supplies  an  engine  with  steam,  having  a  64-inch 
cylinder,  and  with  the  piston  travelling  220  feet  per  minute? 

Here  64  x  64  —  4,096,  which  multiplied  by  220  =  901,120, 
and  this  divided  by  14,000  =  64'3,  which  is  the  proper  area  of 
the  safety  valve  in  square  inches. 

ANOTHER   EULE   FOE   SAFETY   VALVES. 

Multiply  the  nominal  horse  power  of  the  engine  ty  '375,  and  to 
the  product  add  16'875.  The  sum  is  the  proper  area  of  the 
safety  valve  in  square  inches,  when  the  boiler  is  low  pressure. 

Example. — What  is  the  proper  diameter  of  the  safety  valve 
for  a  low  pressnre  engine  the  nominal  power  of  which  is  140 
horses  ? 

Here  140  x  '375  =  52 -5,  adding  to  which  the  constant  num- 
ber 16*875,  we  get  69*375,  which  is  the  proper  area  of  the  safety 
valve  in  square  inches  for  a  low  pressure  engine. 

A  60-inch  cylinder  and  6-feet  stroke  is  equal  to  140  nominal 
horses  power,  so  that  this  rule  gives  somewhat  more  than  a 
square  inch  of  area  in  the  valve  for  each  inch  of  diameter  in  the 
cylinder  in  that  particular  size  of  engine. 

The  opening  through  the  safety  valve  must  be  understood  to 
be  the  effective  opening  clear  of  bridges  or  other  obstacles,  and 
the  area  to  be  computed  is  the  area  of  the  smallest  diameter  of 
the  valve.  Most  safety  valves  are  made  with  a  chamfered  edge, 
which  edge  constitutes  the  steam  tight  surface,  and  the  effective 
area  is  what  corresponds  to  the  smaller  diameter  of  the  valve 
and  not  to  the  larger.  All  boilers  should  have  an  extra  or  ad- 
ditional safety  valve  of  the  same  capacity  as  the  other,  which 
may  act  in  case  of  accident  to  the  first  from  getting  jammed  or 
otherwise.  The  dimensions  of  safety  valve  here  computed  is 
that  adequate  for  letting  off  all  the  steam.  But  in  some  cases 
the  whole  steam  is  not  supplied  from  one  boiler,  and  a  safety 
valve  in  such  case  must  be  put  on  each  boiler,  but  of  a  less  area, 


PEOPEK   DIAMETER   OF   THE    FEED   PIPE.  221 

in  proportion  to  the  smaller  volume  of  steam  it  has  to  let  off. 
If  there  are  two  boilers,  the  safety  valve  on  each  will  be  half 
the  area  of  the  foregoing ;  if  three  boilers,  one-third  of  the  area ; 
if  four  boilers,  one-fourth  of  the  area ;  and  so  of  all  other  pro- 
portions. The  area  of  the  waste  steam  pipe  should  be  the  same 
as  that  of  the  safety  valve. 

TO  FIND  THE  PROPER  DIAMETER  OF  THE  FEED  POPE. 

EULE. — Multiply  the  nominal  horse  power  of  the  engine  as  com- 
puted by  the  Admiralty  rule  by  '04,  to  the  product  add  3  ; 
extract  the  square  root  of  the  sum.  The  result  is  the  diam- 
eter of  the  feedpipe  in  inches. 

Example  1. — "What  is  the  proper  diameter  of  the  feed  pipe  in 
inches  of  an  engine  whose  nominal  horse  power  is  140  ? 

140  =  nominal  horse  power  of  engine 
•04  =  constant  multiplier 


5-6 

3     =  constant  to  be  added 


8-6 


and  v8'6  =  2'93  diameter  of  feed  pipe  in  inches. 


Example  2. — What  is  the  proper  diameter  of  the  feed  pipe 
in  inches  in  the  case  of  an  engine  whose  nominal  horse  power 
is  385  ? 

385  =  nominal  horse  power  of  engine 
•04  =  constant  multiplier 

15-4 
3    constant  to  be  added 


18-4 


and  4/18'4  =  4'29  diameter  of  feed  pipe  in  inches. 


222  PROPORTIONS   OF   STEAM-ENGINES. 

TO   FIND  THE  PROPER  DIMENSIONS  OF  THE  AIR  PUMP  AND 
CONDENSER. 

In  land  engines  the  diameter  of  the  air  pump  is  made  half 
that  of  the  cylinder,  and  the  length  of  stroke  half  that  of  the 
cylinder,  so  that  the  capacity  is  -Jth  that  of  the  cylinder ;  and 
the  condenser  is  made  of  the  same  capacity.  But  in  marine  en- 
gines the  diameter  of  the  air  pump  is  made  '6  of  the  diameter 
of  the  cylinder,  and  the  length  of  the  stroke  is  made  from  '57  to 
•6  times  the  stroke  of  the  cylinder,  and  the  condenser  is  made 
at  least  as  large.  In  some  cases  the  air  pump  is  now  made  dou- 
ble-acting, in  which  case  its  capacity  need  only  he  half  as  great 
as  when  made  single-acting. 

TO  FIND  THE  PROPER  AREA  OF  THE  INJECTION  PIPE. 

RULE. — Multiply  the  nominal  horsepower  of  the  engine,  as  com- 
puted ty  the  Admiralty  rule,  Tjy  0'69,  and  to  the  product 
add  2481.  The  sum  is  the  proper  area  of  the  injection  pipe 
in  square  inches. 

Example  1. — What  is  the  proper  area  of  the  injection  pipe  in 
square  inches  of  an  engine  whose  nominal  horse  power  is  140  ? 

140  =  nominal  horse  power  of  engine 
•069  =  constant  multiplier 

9-66 

2-81  =  constant  to  be  added 


Answer  12-47  =  area  of  injection  pipe  in  square  inches. 

^Example  2. — What  is  the  proper  area  of  the  injection  pipe  in 
square  inches  of  an  engine  whose  nominal  horse  power  is  385  ? 

385  =  nominal  horse  power  of  engine 
•069  =  constant  multiplier 

26-56 
2-81  =  constant  to  be  added 


Answer   29-37  =  area  of  injection  pipe  in  square  inches. 


PROPER  AREA  OP  THE  FOOT  VALVE  PASSAGE.   223 

The  area  of  the  injection  orifice  is  usually  made  about  l-250th 
part  of  the  area  of  the  piston,  which,  in.  an  engine  of  385  horse 
power,  would  be  about  27*7  inches  of  area.  For  warm  climates 
the  area  should  be  increased. 

TO  FIND  THE  PROPER  AREA  OF  THE  FOOT  VALVE  PASSAGE. 

KUIE. — Multiply  the  nominal  horse  power  of  the  engine  ~by  9, 
divide  the  product  T>y  5,  add  8  to  the  quotient.  The  sum  is 
the  proper  area  of  foot  valve  passage  in  square  inches. 

Example  1. — What  is  the  proper  area  of  the  foot  valve  pas- 
sage in  square  inches  of  an  engine  whose  nominal  horse  power 
is  140? 

140  =  nominal  horse  power  of  engine 
9  =  constant  multiplier 


constant  divisor  5)1260 


252 
8  =  constant  to  be  added 


Answer      260  —  area  of  foot  valve  passage  in  square  inches. 


Example  2. — What  is  the  area  of  foot  valve  passage  in  square 
inches  of  an  engine  whose  nominal  horse  power  is  385  ? 

385  —  nominal  horse  power  of  engine 
9  =  constant  multiplier 


constant  divisor  5)3465 


693 
8  =  constant  to  be  added 


Answer      701  —  area  of  foot  valve  passage  in  square  inches. 

The  discharge  valve  passage  is  made  of  the  same  size  as  the 
foot  valve  passage. 

A  common  rule  for  the  area  of  the  foot  and  discharge  valve 
passages  is  one-fourth  of  the  area  of  the  air  pump,  and  the  waste 


224  PROPORTIONS    OF   STEAM-ENGINES. 

water  pipe  is  made  one-fourth  of  the  diameter  of  the  cylinder, 
which  gives  a  somewhat  less  area  than  that  through  the  foot  and 
discharge  valve  passages.  Such  rules,  however,  are  only  appli- 
cable to  slow-going  engines.  In  rapid-working  engines,  such  as 
those  employed  for  driving  the  screw  propeller  by  direct  action, 
and  in  which  the  air-pump  is  usually  double  acting,  the  area 
through  the  foot  and  discharge  valves  should  be  equal  to  the 
area  of  the  air-pump,  and  the  waste  water  pipe  should  also  have 
the  same  area.  In  all  cases,  therefore,  in  which  these  or  other 
rules  dependent  on  the  nominal  power  are  applied  to  fast-going 
engines,  the  nominal  power  must  be  computed  by  the  Admiralty 
rule,  in  which  the  speed  of  the  piston  is  taken  into  account. 

TO  FIND  THE  PKOPEE  DIAMETER  OP  THE  WASTE  WATER 
PIPE. 

RULE. — Multiply  the  square  root  of  the  nominal  horsepower  of 
the  engine  ty  1/2.  The  product  is  the  diameter  of  the  waste 
water  pipe  in  inches. 

Example  1. — What  is  the  diameter  of  the  waste  water  pipe, 
in  inches,  of  an  engine  whose  nominal  horse  power  is  140  ? 

140  =  nominal  horse  power  of  engine 
and  |/140=11.83 

1'2    =  constant  multiplier 

Answer   1419  =  diameter  of  waste  water  pipe  in  inches. 


Example  2. — What  is  the  diameter  of  waste  water  pipe,  in 
inches,  of  an  engine  whose  nominal  horse  power  is  385  ? 

386  =  nominal  horse  power  of  engine 
and  y  385  =  19-62 

1-2   =  constant  multiplier 

Answer   23'64  =  diameter  of  waste  water  pipe  in  inches. 


CAPACITY  OF  THE  FEED  PUMP. 

The  relative  volumes  of  steam  and  water  are  at  15  Ibs.  on  the 
square  inch,  or  the  atmospheric  pressure,  1,669  to  1 ;  at  SO  Ibs.,  or 


PROPER    DIMENSIONS    OF   THE    FEED   PUMP.  225 

15  Ibs.  on  the  square  inch  above  the  atmospheric  pressure,  881  to 
1 ;  at  60  Ibs.,  or  45  Ibs.  above  the  atmospheric  pressure,  467  to  1 ; 
and  at  120  Ibs.,  or  105  Ibs.  above  the  atmospheric  pressure,  249  to  1. 
In  every  engine,  taking  into  account  the  risks  of  leakage  and 
priming  in  the  boiler,  the  feed  pump  should  be  capable  of  dis- 
charging twice  the  quantity  of  water  that  is  consumed  in  the 
generation  of  steam ;  and  in  marine  boilers  it  is  necessary  to 
blow  out  as  much  of  the  supersalted  water  as  the  quantity  that 
is  raised  into  steam,  in  order  to  keep  the  boiler  free  from  saline 
incrustations.  But  if  this  water  is  discharged  by  leakage  or 
priming,  the  object  of  preventing  salting  is  equally  fulfilled. 
Pumps,  especially  if  worked  at  a  high  rate  of  speed,  do  not  fill 
themselves  with  water  at  each  stroke,  but  sometimes  only  half 
fill  themselves,  and  sometimes  do  not  even  do  that.  Then  in 
steam  vessels,  one  pump  should  be  able  to  supply  both  engines 
with  steam,  and  the  pump  is  generally  only  single-acting,  while 
the  cylinder  is  double-acting.  If,  therefore,  we  wish  to  see 
what  size  of  pump  we  ought  to  supply  to  an  engine  in  which 
the  terminal  elasticity  of  steam  in  the  cylinder  is  equal  to  the 
atmospheric  pressure,  we  know  that  the  quantity  of  water  in 
the  steam  is  just  -j-^g-jth  of  the  volume  of  the  steam ;  but.  as  we 
require  to  double  the  supply  to  make  up  for  waste,  the  volume 
of  water  supplied  will  on  this  ground  be  y^-y ;  and  as  the  pump 
may  only  half  fill  itself  every  stroke,  the  capacity  of  the  pump 
must  on  this  ground  be  1-/6-5-  of  the  volume  of  steam.  But  then 
the  pump  is  only  single-acting,  while  the  cylinder  is  double-act- 
ing, on  which  account  the  capacity  of  the  pump  must  be  doubled, 
in  order  that  it  may  in  a  half  stroke  discharge  the  water  re- 
quired to  produce  the  steam  consumed  in  a  whole  stroke.  This 
would  make  the  capacity  of  the  pump  T-&-J,  or  ^fa  of  the  capa- 
city of  the  cylinder,  and  a  less  proportion  than  this  is  inadvisa- 
ble in  the  case  of  marine  engines.  Even  with  this  proportion, 
one  feed  pump  would  not  supply  all  the  boilers,  as  it  ought  to 
be  able  to  do  in  case  of  accident  happening  to  the  other,  unless 
it  should  happen  that  the  pump  draws  itself  full  of  water  at 
each  stroke  instead  of  half  full,  as  it  will  nearly  do  if  the  mo- 
tion of  the  engine  is  slow  and  the  passages  leading  into  it  large, 


226  PROPORTIONS    OF   STEAM-ENGINES. 

and  if  at  the  same  time  tlie  valves  are  large  and  have  not  much 
lift.  In  the  case  of  engines  working  at  a  high  speed,  ^-jro  of  the 
capacity  of  the  cylinder  for  the  capacity  of  the  feed  pump  is 
scarcely  sufficient,  especially  if  there  be  no  air  vessel  on  the 
suction  side  of  the  pump,  which  in  such  pumps  should  always 
be  introduced.  In  the  engines  of  the  'Clyde,'  'Tweed,'  'Tay,' 
and  '  Teviot,'  by  Messrs.  Caird,  the  feed  pump  is  ^yth  of  the 
capacity  of  the  cylinder.  In  steam  vessels  there  is  no  doubt 
always  the  resource  of  the  donkey  engine  to  make  up  for  any 
deficiency  in  the  feed.  But  it  is  much  better  to  have  the  main 
feed  pumps  of  the  engine  made  of  sufficient  size  to  compensate 
for  all  the  usual  accidents  befalling  the  supply  of  feed  water. 
Of  course,  the  supply  of  feed  water  required  will  vary  mate- 
rially with  the  amount  of  expansion  with  which  the  steam  is 
worked,  and  also  with  the  amount  of  superheating ;  and  in  the 
old  flue  boilers  with  the  chimney  passing  up  through  the  steam 
chest,  there  was  always  a  considerable  degree  of  superheating. 
A  rule  applicable  to  all  pressures  of  steam  and  to  moderate  rates 
of  expansion  is  as  follows : — 

TO   FIND   THE   PEOPEE   CAPACITY   OF   THE   FEED   PUMP. 

RULE. — Multiply  the  capacity  of  the  cylinder  in  cubic  inches  by 
the  total  pressure  of  the  steam  in  the  toiler  on  each  square 
inch  (or  <by  the  load  on  each  square  inch  of  the  safety  valve 
plus  15  Ibs.  on  each  square  inch  for  the  pressure  of  the  at- 
mosphere), and  divide  the  product  ly  4,000.  The  quotient 
is  the  proper  capacity  of  the  feed  pump  in  cubic  inches  when 
the  pump  is  single-acting  and  the  engine  is  double-acting. 

If  the  pump  should  be  double-acting,  one-half  of  the  above 
capacity  will  suffice. 

Example  1. — "What  is  the  proper  volume  of  the  working  part 
of  the  plunger  of  an  engine  with  a  74-inch  cylinder  and  7^-feet 
stroke,  the  steam  in  the  boiler  being  5  Ibs.  per  square  inch  above 
the  atmospheric  pressure  ? 

The  area  in  square  inches  of  a  circle  74  inches  diameter  is 
4,300,  which,  multiplied  by  7i  feet  or  90  inches,  gives  387,000 


COLD   WATER   PUMP.  227 

cubic  inches  as  the  capacity  of  the  cylinder.  Now  if  the  steam 
in  the  boiler  be  5  Ibs.  per  square  inch  above  the  atmosphere,  it 
will  have  a  total  pressure  of  5  +  15,  or  20  Ibs.  per  square  inch. 
Multiplying,  therefore,  387,000  by  20,  we  get  7,740,000,  which, 
divided  by  4,000,  gives  1,935  as  the  proper  capacity  of  the  feed 
pump  in  cubic  inches.  If  now  the  stroke  of  the  pump  be  51 
inches,  we  divide  1,935  by  51,  which  gives  us  38  inches  nearly 
as  the  proper  area  of  the  feed  pump  plunger.  This  area  corre- 
sponds to  a  diameter  of  7  inches,  which  is  a  better  proportion 
than  that  subsisting  in  the  engines  of  the  '  Clyde,'  '  Tweed,' 
'  Tay,'  and  '  Teviot,'  which,  with  a  74-inch  cylinder,  7i  feet 
stroke,  and  51  inches  stroke  of  pump,  had  the  feed  pump  plung- 
ers of  only  6  inches  diameter. 

Example  2. — What  is  the  proper  volume  of  the  working  part 
of  the  plunger  of  a  locomotive  feed  pump,  having  cylinders  of 
18  inches  diameter  and  2  feet  stroke,  working  with  a  pressure 
of  85  Ibs.  pressure  above  the  atmosphere  ? 

The  area  of  a  circle  18  inches  diameter  is  254-5  square  inches, 
which,  multiplied  by  24  inches,  which  is  the  length  of  the 
stroke,  gives  6,108  cubic  inches  as  the  capacity  of  the  cylinder. 
If  the  steam  be  85  Ibs.  above  the  atmosphere,  then  the  total  press- 
ure must  be  100  Ibs.  per  square  inch,  and  6,108  x  100=610,800, 
which,  divided  by  4,000,  gives  152'7  as  the  capacity  of  the  feed 
pump  in  cubic  inches.  This  is  a  somewhat  larger  proportion  of 
feed  pump  than  is  usually  given  in  locomotive  engines.  In  the 
locomotive  '  Iron  Duke '  the  diameter  of  the  feed  pump  plunger 
is  2J  inches  and  the  stroke  24  inches.  But  152'7  divided  by  24 
inches  gives  an  area  of  6'36  square  inches,  which  answers  to  a 
diameter  of  plunger  of  2£  inches.  In 'locomotives,  however,  as 
in  marine  engines,  the  feed  pumps  are  very  generally  made  too 
small,  so  that  the  proportion  given  in  the  rule  appears  prefera- 
ble to  that  commonly  adopted. 

COLD-WATER  PUMP. 

The  proper  dimensions  of  the  cold-water  pump  can  easily  be 
determined  by  a  reference  to  the  number  of  cubic  inches  of  wa- 
ter, at  a  given  temperature,  that  are  required  to  condense  a 


223  PROPORTIONS    OF   STEAM-ENGINES. 

cubic  inch  in  the  form  of  steam.  There  is  no  need,  however, 
of  going  through  the  details  of  the  process,  and  the  proper  di- 
mensions of  the  pump  will  be  found  by  the  following  rule : — 

TO  DETERMINE   THE   PBOPEE  DIMENSIONS   OF  THE   OOLD-WATEB 
PTJMP. 

RULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  ~by  the  length  of  the  stroke  in  feet,  and  divide  the 
'product  l>y  4,400.  The  quotient  is  the  proper  capacity  of 
the  cold-water  pump  in  cubic  feet. 

Example  1. — "What  is  the  proper  capacity  of  the  cold-water 
pump  in  an  engine,  having  a  60-inch  cylinder  and  a  5J-feet 
stroke  ? 

Here  60  x  60  —  3,600,  which  multiplied  by  5£  is  19,800,  and 
this  divided  by  4,400  is  4'5,  which  is  the  proper  capacity  of  the 
cold-water  pump  in  cubic  feet. 

Example  2. — "What  is  the  proper  capacity  of  the  cold-water 
pump  hi  the  case  of  an  engine,  with  a  2-feet  cylinder  and  3-feet 
stroke  ? 

Here  24  x  24  =  576,  and  this  multiplied  by  3  =  1,728,  which 
divided  by  4,400  =  '39  cubic  feet,  or  multiplying  -39  by  1,728, 
we  get  the  capacity  in  cubic  inches,  which  is  673'92.  This  is  a 
somewhat  larger  content  than  is  sometimes  given  in  practice. 
Maudslay's  16-horse  land  engine  has  a  24-inch  cylinder  and 
3-feet  stroke,  and  the  cold-water  pump  has  a  diameter  of  6| 
inches,  and  a  stroke  of  18  inches,  which  gives  a  capacity  of  594 
cubic  inches,  instead  of  673,  as  specified  above.  The  larger  di- 
mension is  the  one  to  be  preferred. 

FLY-WHEEL. 

Boulton  and  "Watt's  rule  for  finding  the  sectional  area  of  the 
fly-wheel  rim  is  as  follows : — 

RULE. — Multiply  44,000  times  the  length  of  the  strolce  in  feet 
by  the  square  of  the  diameter  of  the  cylinder  in  inches,  and 
divide  the  product  by  the  square  of  the  number  of  revolu- 
tions per  minute,  multiplied  by  the  cube  of  the  diameter  of 


PROPER   DIMENSIONS    OF   THE    FLY-WHEEL.  229 

the  fly-wheel  in  feet.     The  resulting  number  will  be  the 
proper  sectional  area  of  the  fly -wheel  rim  in  square  inches. 

Example. — What  will  be  the  proper  sectional  area  of  the 
fly-wheel  rim  in  square  inches  in  the  case  of  an  engine,  with  a 
cylinder  24  inches  diameter  and  5  feet  stroke,  the  fly-wheel  he- 
ing  20  feet  diameter. 

Here  44,000  multiplied  by  5,  which  is  the  length  of  the 
stroke  in  feet,  is  220,000.  The  square  of  the  diameter  of  the 
cylinder  in  inches  is  576,  and  220,000  x  576  =126,720,000.  The 
engine  will  make  about  21  revolutions,  the  square  of  which  is 
441,  and  the  cube  of  the  diameter  of  the  fly-wheel  in  feet  is 
8,000,  which  multiplied  by  441  is  3,528,000.  Finally  126,720,000 
divided  by  3,528,000  is  35'8,  which  is  the  proper  area  in  square 
inches  of  the  section  of  the  fly-wheel  rim. 

In  an  engine  constructed  by  Mr.  Oaird,  with  a  24-inch  cylin- 
der, 5-feet  stroke,  and  20-foot  fly-wheel,  the  width  of  the  rim 
was  10  inches,  and  the  thickness  3f  inches,  giving  a  sectional 
area  of  37*5  square  inches,  which  is  somewhat  larger  than  Boul- 
ton  and  "Watt's  proportion. 

Suppose  that  we  take  the  sectional  area  in  round  numbers  at 
36  square  inches,  and  the  circumference  of  the  fly-wheel  or 
length  of  rim  if  opened  out  at  62  feet  or  744  inches,  then  there 
will  be  36  times  744,  or  26,784  cubic  inches  of  cast  iron  in  the 
rim,  or  dividing  by  1,728,  we  shall  have  15-5  cubic  feet  of  cast 
iron.  But  a  cubic  foot  of  cast  iron  weighs  444  Ibs.  Hence  15J 
cubic  feet  will  weigh  6,882  Ibs.,  and  this  weight  revolves  with  a 
speed  of  21  times  62,  or  1,303  feet  per  minute,  or  21-7  feet  per 
second,  or  260-4  inches  per  second.  To  find  the  height  in 
inches  from  which  a  body  must  have  fallen,  to  acquire  any  given 
velocity  in  inches  per  second,  we  square  the  velocity  in  inches, 
and  divide  the  square  by  772*84,  which  gives  the  height  in 
inches.  Now  the  square  of  260*4  is  67,808,  which  divided  by 
772-84  =  87  inches,  or  7J  feet,  so  that  the  energy  treasured  in 
the  fly-wheel  is  equal  to  a  weight  of  6,882  Ibs.  falling  through 
7J  feet,  or  to  a  weight  of  49,984-5  Ibs.  falling  through  1  foot. 
Now  the  area  of  the  cylinder  being  in  round  numbers  452 
square  inches,  the  total  pressure  upon  it,  if  we  allow  an  effec- 


230  PROPORTIONS   OF   STEAM-ENGINES. 

tive  pressure  including  steam  and  vacuum  of  7  Ibs.  per  square 
inch,  as  was  the  proportion  allowed  in  Watt's  engines,  will  be 
3,164  Ibs.,  and  the  length  of  stroke  being  5  feet,  we  shall  have 
3,164  Ibs.  moved  through  5  feet,  or  5  times  this,  which  is  15,820 
Ibs.  moved  through  1  foot  in  each  half  stroke  of  the  engine. 
Dividing  now  49,984*5  foot-pounds,  the  total  power  resident  in 
the  fly-wheel  at  its  mean  velocity,  by  158'20  foot-pounds,  which 
is  the  power  developed  in  each  half  stroke  of  the  engine,  we 
get  3'1  as  the  resulting  number,  which  shows  that  there  is  over 
three  times  the  power  resident  in  the  fly-wheel  that  is  devel- 
oped in  each  half  stroke  of  the  engine.  In  cases  where  great 
equability  of  motion  is  required,  this  power  of  fly-wheel  is  not 
sufficient,  and  in  some  engines,  the  proportion  is  made  six  times 
the  power  developed  in  each  half  stroke,  or,  in  other  words,  the 
fly-wheel  is  twice  as  heavy  as  that  computed  above. 

GOVERNOR. 

The  altitude  of  the  height  of  the  cone  in  which  the  arms  re- 
volve, measuring  from  the  plane  of  revolution  to  the  centre  of 
suspension,  will  be  the  same  as  that  of  a  pendulum  which  makes 
the  same  number  of  double  beats  per  minute  that  the  governor 
makes  of  revolutions ;  or  if  the  number  of  revolutions  per  minute 
be  fixed,  and  we  wish  to  obtain  the  proper  height  of  cone,  we 
divide  the  constant  number  375'36  by  twice  the  number  of  revo- 
lutions, which  gives  the  square  root  of  the  height  of  the  cone ; 
and,  consequently,  the  height  itself  is  equal  to  the  square  of  this 
number.  These  relations  are  exhibited  in  the  following  rules : — 

TO  DETERMINE  THE  8PEED  AT  WHICH  A  GOVERNOR  MUST  BE 
DRIVEN,  WHEN  THE  HEIGHT  OF  THE  CONE  18  FIXED  IN  WHICH 
THE  ARMS  REVOLVE. 

RULE. — Divide  the  constant  number  375'36  ~by  twice  the  square 
root  of  the  height  of  the  cone  in  inches.  The  quotient  is  the 
proper  number  of  revolutions  per  minute. 

Example. — A  governor  with  arms  30£  inches  long,  measuring 
from  the  centre  of  suspension  to  the  centre  of  the  ball,  revolves 


PROPER   PROPORTIONS   OF   THE    GOVERNOR.  231 

in  the  mean  position  of  the  arms  at  an  angle  of  about  30  degrees, 
with  the  vertical  spindle  forming  a  cone  about  26J  inches  high. 
At  what  number  of  revolutions  per  minute  should  this  governor 
be  driven  ? 

Here  the  height  of  the  cone  being  26-5  inches,  the  square  root 
of  which  is  5*14,  and  twice  the  square  root  10'28,  we  divide 
375-36  by  10-28,  which  gives  us  36-5  as  the  proper  number  of 
revolutions  at  which  the  governor  should  be  driven. 

TO  DETERMINE  THE  HEIGHT  OF  THE  CONE  IN  WHICH  THE  AEM3 
MUST  EEVOLVE,  WHEN  THE  VELOCITY  OF  EOTATION  OF  THE 
GOVEENOE  IS  DETEBMINED. 

ETJLE. — Divide  the  constant  number  375-36  ly  twice  the  number 
of  revolutions  which  the  governor  makes  per  minute,  and 
square  the  quotient,  which  will  fie  the  height  in  inches  which 
the  cone  will  assume. 

Example. — Suppose  that  a  governor  be  driven  with  a  speed 
of  36£  revolutions  per  minute,  what  will  be  the  height  of  the 
cone  in  which  the  balls  will  necessarily  revolve,  measuring  from 
the  centre  of  suspension  of  the  arms  to  the  plane  of  revolution 
of  the  balls? 

Here  36-5  x  2  =  73,  and  375-36  divided  by  73  =  5-14,  and 
5*14  squared  is  equal  to  26-4196,  or  very  nearly  26-5  inches, 
which  will  be  the  height  of  the  cone. 

When  the  arms  revolve  at  an  angle  of  45  degrees  with  the 
spindle,  or  at  right  angles  with  one  another,  the  centrifugal  force 
is  equal  to  the  weight  of  the  balls ;  and  when  the  arms  revolve 
at  an  angle  of  30  degrees  with  the  spindle,  they  form  with  the 
base  of  the  cone  an  equilateral  triangle. 


STRENGTHS    OF   LOW-PEESSUEE    LAND    ENGINES. 

PISTON    BOD. 

The  piston  rod  is  made  one-tenth  of  the  diameter  of  the 
cylinder,  except  in  locomotives,  where  it  is  made  one-seventh 


232  PROPORTIONS    OF    STEAM-EXGINES. 

of  the  diameter.  The  piston  rod  is  sometimes  made  of  steel,  or 
of  iron  converted  into  steel  for  a  certain  depth  in.  This  enables 
it  to  acquire  and  maintain  a  better  polish  than  if  made  of  iron. 

MAIN  LINKS. 

The  main  links  are  the  parts  which  connect  the  piston  rod 
with  the  beam.  They  are  usually  made  half  the  length  of  the 
stroke,  and  their  sectional  area  is  113th  the  area  of  the  piston. 

AIR-PUMP   ROD. 

The  diameter  of  the  air-pump  rod  is  commonly  made  one- 
tenth  of  the  diameter  of  the  air-pump. 

BACK  LINKS. 

The  sectional  area  of  the  back  links  is  made  the  same  as  that 
of  the  air-pump  rod. 

END   STUDS  OF   THE    BEAM. 

The  end  studs  of  the  beam  are  usually  made  the  same  diam- 
eter as  the  piston  rod.  Sometimes  they  are  of  cast-iron,  but 
generally  now  of  wrought.  The  gudgeons  of  water  wheels  are 
generally  loaded  with  about  500  Ibs.  for  every  circular  inch  of 
their  transverse  section,  which  is  nearly  the  proportion  that  ob- 
tains in  the  end  studs  of  engine  beams.  But  the  main  centre  is 
usually  loaded  beyond  this  proportion. 

MAIN   CENTRE. 

The  strength  of  this  part  will  be  given  in  the  strengths  of 
marine  engines.  But  when  of  cast-iron  it  is  usually  made  about 
one-fifth  of  the  diameter  of  the  cylinder. 

In  a  cylinder  of  24  inches  diameter  this  will  be  4'8  inches,  or 
say  4£  inches ;  and  this  proportion  of  strength  will  be  about  nine 
times  the  breaking  weight,  if  we  suppose  the  main  centre  to  be 
overhung  as  in  marine  engines.  Thus,  in  a  cylinder  of  24  inches 
diameter,  and,  consequently,  of  452  square  inches  area,  the  total 
load  on  the  piston  with  20  Ibs.  on  each  square  inch  is  9,040  Ibs. 


PEOPER   DIMENSIONS    OF   THE   MAIN   BEAM.  233 

But  as  the  strain  at  the  main  centre  is  doubled  from  the  beam 
acting  as  a  lever  of  2  to  1,  it  follows  that  the  strain  at  the  main 
centre  will  be  18,080  Ibs.  The  ultimate  tensile  strength  of  com- 
mon cast-iron  being  12,000  per  square  inch  of  section,  and  the 
tensile  and  shearing  strength  being  about  the  same,  ^th  of  12,000, 
or  1,333  Ibs.,  will  be  the  proper  load  to  place  on  each  square  inch 
of  section  ;  and  18,080  divided  by  1,333  will  give  the  proper  sec- 
tional area  in  square  inches,  which  will  be  13$-  square  inches 
nearly.  This  area  corresponds  to  a  diameter  of  a  little  over  Cl- 
inches. But  the  strength  is  virtually  doubled  by  the  circum- 
stance of  the  main  centre  of  land  engines  being  supported  at 
both  ends. 

MAIN  BEAM. 

The  rules  in  common  use  for  proportioning  the  main  beams 
of  engines  are  the  same  as  those  which  existed  prior  to  Mr. 
Hodgkinson's  researches  on  the  strength  of  cast-iron  girders, 
which  showed  that  the  main  element  of  strength  was  the  bot- 
tom flange.  But  as  in  the  case  of  engine  beams  the  strain  is 
alternately  up  and  down,  the  top  and  bottom  flanges,  or  beads 
of  the  beam,  require  to  be  of  equal  strength.  Cast-iron  is  a  bad 
material  for  engine  beams,  unless  the  central  part  be  made  of 
open  work  of  cast-iron,  and  the  edge  of  the  beam  be  encircled 
by  a  great  elliptical  or  lozenge-formed  hoop,  as  is  done  in  some 
of  the  American  engines.  But  if  the  beam  be  made  wholly  of 
cast-iron,  a  much  larger  proportion  of  the  metal  should  be  col- 
lected in  the  top  and  bottom  flanges  than  is  at  present  the  ordi- 
nary practice. 

The  usual  length  of  the  main  beam  is  three  times  the  length 
of  the  stroke ;  the  usual  breadth  is  equal  to  the  diameter  of  the 
cylinder,  and  the  usual  mean  thickness  is  yj^th  of  the  length. 
The  rule  is  as  follows : — 

TO     FIND    THE    PBOPEK    DIMENSIONS    OP    THE     MAIN    BEAM    OF   A 
LAND   ENGINE. 

RULE. — Divide  the  weight  in  Ibs.  acting  at  the  centre  Try  250  and 
multiply  the  quotient  by  the  distance  between  the  extreme  cen- 
tres. To  find  the  depth,  the  breadth  being  given:  Divide  the 


234  PROPORTIONS    OF    STEAM-ENGINES. 

product  ~by  the  breadth  in  indies,  and  extract  the  square  root 

of  the  quotient,  which  is  the  depth. 

The  depth  of  the  beam  at  the  ends  is  usually  made  one-third 
of  the  depth  at  the  middle. 

It  will  be  preferable,  however,  to  investigate  a  rule  on  the 
basis  of  Mr.  Hodgkinson's  rule  for  proportioning  cast-iron  gird- 
ers, which  is  as  follows : 

Multiply  the  sectional  area  of  the  bottom  flange  in  inches  by 
the  depth  of  the  beam  in  inches,  and  divide  the  product  by  the 
distance  between  the  supports  also  in  inches,  and  514  times  the 
quotient  will  le  the  breaking  weight  in  cwts. 

If  the  breaking  weight  be  expressed  in  tons,  the  constant 
number  514  must  be  divided  by  20,  which  gives  the  breaking 
weight  as  25- 7,  or  say  26  tons,  whereas  experiment  has  shown 
that  if  the  flange  were  to  be  formed  of  malleable  iron  instead 
of  cast,  the  breaking  weight  would  not  be  less  than  80  tons ;  or, 
in  other  words,  that  with  the  same  sectional  area  of  flange,  the 
beam  would  be  more  than  three  times  stronger. 

It  is  a  common  practice  in  the  case  of  girders  to  make  the 
strength  equal  to  three  times  the  breaking  weight  when  the  load 
is  stationary,  and  to  six  times  the  breaking  weight  when  the 
load  is  movable.  But  these  proportions  are  too  small,  and  less 
than  nine  or  ten  times  the  breaking  weight  will  not  give  a  suf- 
ficient margin  of  strength  in  the  case  of  engines  where  the  mo- 
tion is  so  incessant,  and  where  heavy  strains  may  be  accidentally 
encountered  from  priming  or  otherwise.  In  the  case  of  an  en- 
gine, the  weight  answering  to  the  breaking  weight  is  the  load 
on  the  piston ;  and  if  we  suppose  the  fly-wheel  to  be  jammed, 
and  the  piston  to  be  acting  with  its  full  force  to  lift  or  sink  the 
main  centre,  it  is  clear  that  the  strain  on  the  main  centre,  and, 
therefore,  on  the  beam,  will  be  equal  to  twice  the  strain  upon 
the  piston,  since  the  beam  acts  under  such  circumstances  as  a 
lever  of  2  to  1.  The  problem  we  have  now  to  consider  is  how 
many  times  the  working  weight  must  be  less  than  the  breaking 
weight  to  give  a  sufficient  margin  of  strength  in  any  given  beam ; 
or,  in  other  words,  what  proportions  must  the  beam  have  to 
possess  adequate  working  strength. 


PROPER    DIMENSIONS    OF  THE    MAIN   BEAM.  235 

To  take  a  practical  example  from  an  engine  in  constant  work. 
The  engine  with  a  cylinder  of  24  inches  diameter  has  a  main 
beam  15  feet  (or  180  inches)  long ;  30  inches  deep  in  the  middle ; 
and  with  a  sectional  area  of  flange  of  7  square  inches..  The 
breaking  weight  of  such  a  beam  in  cwts.  will  be  7x30x514 
divided  by  180=600  cwt.  nearly,  and  this  multiplied  by  112  Ibs. 
=  67,200  lb.,  which  is  the  breaking  weight  in  pounds  avoirdu- 
pois. The  area  of  the  cylinder  in  round  numbers  is  452  square 
inches ;  but  as  there  is  a  leverage  of  2  to  1,  this  is  equivalent  to 
an  area  of  cylinder  of  904  square  inches  set  under  the  middle  of 
the  beam  and  pulling  it  downwards,  the  beam  being  supposed 
to  be  supported  at  both  ends.  Dividing  now  67,200  by  904  we 
get  the  pressure  per  square  inch  on  the  piston  that  would  break 
the  beam,  which  is  a  little  over  74  Ibs.  per  square  inch  of  the 
area  of  the  piston,  or  58  Ibs.  per  circular  inch.  If  we  suppose 
the  working  pressure  of  steam  on  the  piston  to  be  6'27  Ibs.  per 
circular  inch,  or  7'854  Ibs.  per  square  inch,  then  the  working 
strength  of  the  beam  will  be  about  9£  times  its  breaking  strength, 
which  would  give  an  adequate  margin  for  safety.  But  if  we 
suppose  the  working  pressure  to  be  12 '54  Ibs.  per  circular  inch, 
or  15'718  Ibs.  per  square  inch,  the  working  strength  would  in 
such  case  be  only  about  4£  times  the  breaking  strength,  and  the 
beam  would  be  too  weak. 

The  strength  of  a  cast-iron  beam  of  any  given  dimensions 
varies  directly  as  the  sectional  area  of  the  edge  flange ;  or,  if 
the  sectional  area  of  that  flange  be  constant,  the  strength  of  the 
beam  varies  directly  as  the  depth,  and  inversely  as  the  length. 
If  while  the  sectional  area  of  the  flange  remains  the  same  the 
depth  of  the  beam  is  doubled  without  altering  the  length,  then 
the  strength  is  doubled.  But  if  the  length  be  also  doubled,  the 
strength  remains  the  same  as  at  first.  As  the  length  of  an  en- 
gine-beam is  doubled  when  we  double  the  length  of  the  stroke, 
and  as  in  any  symmetrical  increase  of  an  engine  when  we  double 
the  length  of  the  stroke  we  also  double  the  diameter  of  the  cyl- 
inder, to  which  the  depth  of  the  beam  is  generally  made  equal, 
large  beams  with  the  same  area  of  flange,  and  made  in  the  ordi- 
nary proportions,  would  be  as  strong  as  small  beams,  except  that 


236  PROPORTIONS   OF   STEAM-ENGINES. 

the  load  increases  as  the  square  of  the  diameter  of  the  cylinder, 
and  consequently  the  area  of  the  edge  flange  must  increase  in 
the  same  proportion.  These  considerations  enable  us  to  fix  the 
following  rule  for  the  strength  of  main  beams  : — 


TO   FIND   THE   PKOPER   DIMENSIONS    OF   THE   MAIN  BEAM   OF   AN 
ENGINE. 

RULE. — Make  the  depth  of  the  beam  equal  to  the  diameter  of  the 
cylinder,  and  the  length  of  the  beam  equal  to  three  times  the 
length  of  the  stroke.  Then  to  find  the  area  of  the  edge 
flange :  Multiply  the  area  of  the  cylinder  in  square  inches 
l>y  the  total  pressure  of  steam  and  vacuum  on  each  square 
inch  of  the  piston,  and  divide  the  product  l>y  650.  The 
quotient  is  the  proper  area  of  the  flange  of  the  beam  in 
square  inches. 

Example  1.— "What  is  the  proper  sectional  area  of  the  flange 
of  the  main  beam  of  an  engine,  with  cylinder  24  inches  diam- 
eter and  5-feet  stroke,  the  pressure  on  the  piston  being  20  Ibs. 
per  square  inch  ? 

Here  the  area  of  the  cylinder  will  be  452  inches,  which  mul- 
tiplied by  20  gives  9,040,  and  dividing  by  650  we  get  13'9  square 
inches,  which  is  the  proper  sectional  area  of  the  edge  bead  or 
flange  of  the  beam. 

JExample  2. — What  is  the  proper  sectional  area  of  the  flange 
of  the  main  beam  of  an  engine  with  a  cylinder  60  inches  di- 
ameter, 12 J  feet  stroke,  and  with  a  pressure  of  steam  on  the  pis- 
ton of  20  Ibs.  per  square  inch  ? 

The  area  of  a  cylinder  60  inches  diameter  is  2,824  square 
inches,  and  2,824  multiplied  by  20=56,480,  which  divided  by 
650=87  square  inches  nearly.  Such  a  flange,  therefore,  if  14£ 
inches  broad,  would  be  6  inches  thick.  The  beam  would  be 
5  feet  deep  at  the  middle,  and  37J  feet  long  between  the  ex- 
treme centres. 


PROPER  DIMENSIONS   OP   THE   MAIN   BEAM.  237 

ANOTHER  ETTLE   FOR  FINDING   THE   SECTIONAL  AREA   OF   EACH 
EDGE  FLANGE   OF  THE  MAIN  BEAM. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  one- 
third  of  the  length  of  the  stroke  in  inches,  and  by  the  total 
pressure  on  each  square  inch  of  the  piston,  and  divide  the 
product  ly  650.  The  quotient  is  the  proper  sectional  area 
in  square  inches  of  each  flange  or  head  on  the  edge  of  the 
"beam. 

Example  1. — What  is  the  proper  sectional  area  of  the  flange 
on  the  edge  of  the  main  beam  of  an  engine  with  a  24-inch  cylin- 
der, 20  Ibs.  total  pressure  on  piston  per  square  inch,  and  5  feet 
stroke  ? 

Here  24  x  20  (which  is  one-third  of  the  stroke  in  inches)  x  20 
(the  pressure  of  the  steam  and  vacuum  per  square  inch)  =  9,600, 
which  divided  by  650=14'7  sq.  in.,  which  is  the  area  required. 

Example  2. — "What  is  the  proper  sectional  area  of  the  flange 
on  the  edge  of  the  main  beam  of  an  engine  with  a  60-inch  cylin- 
der, 12^-feet  stroke,  and  with  a  pressure  on  the  piston  of  20  Ibs. 
per  square  inch  ? 

Here  60  x  50  (which  is  one-third  of  the  stroke  in  inches)  x  20, 
(the  pressure  of  the  steam  per  square  inch)  =  6,000,  which  di- 
vided by  650  gives  92  as  the  sectional  area  of  the  edge  bead  in 
square  inches.  Such  a  flange,  if  15£  inches  broad,  would  be 
6  inches  thick.  These  results  it  will  be  seen  are  very  nearly 
the  same  as  those  obtained  by  the  preceding  rule ;  and  one  in- 
ference from  these  rules  is  that  nearly  all  engine  beams  are  at 
present  made  too  weak.  The  purpose  of  the  web  of  the  beam 
is  mainly  to  connect  together  the  top  and  bottom  flanges,  so  that 
there  is  no  advantage  in  making  it  thicker  than  suffices  to  keep 
the  beam  in  shape ;  with  which  end,  too,  stiffening  feathers,  both 
vertical  and  horizontal,  should  be  introduced  upon  the  sides  of 
the  beam.  The  first  cast-iron  beams  were  made  like  a  long 
hollow  box  to  imitate  wooden  beams,  and  this  form  would  still 
be  the  best,  unless  an  open  or  skeleton  beam,  encircled  with 
a  great  wrought-iron  hoop,  after  the  American  fashion,  be 
adopted. 


238  PROPORTIONS   OF   STEAM-ENGINES. 

CONNECTING-ROD. 

The  connecting-rods  of  land  engines  are  now  usually  made 
of  wrought-iron,  and  when  so  made,  the  proportions  will  be  the 
same,  or  nearly  so,  as  those  given  under  the  head  of  marine  en- 
gines. When  made  of  cast-iron  the  configuration  is  such  that 
the  transverse  section  at  the  middle  assumes  the  form  of  a  cross, 
this  form  being  adopted  to  give  greater  lateral  stiffness.  The 
length  of  the  rod  is  usually  made  the  same  as  the  length  of  the 
beam,  namely,  three  times  the  length  of  the  stroke,  and  the 
area  of  the  cross  section  of  the  rod  at  the  middle  is  commonly 
made  ^th  of  the  area  of  the  cylinder,  and  the  sectional  area  at 
the  ends  ^jth  of  the  area  of  the  cylinder.  Such  a  strength  is 
needlessly  great,  and  is  quite  out  of  proportion  to  the  strength 
commonly  given  to  the  beam.  Thus,  in  the  case  of  an  engine 
with  a  24-inch  cylinder,  the  area  of  the  piston  is  452  square 
inches ;  and  if  we  take  20  Ibs.  per  square  inch  as  the  load  on  the 
piston,  then  the  total  load  on  the  piston  will  be  9,040  Ibs.  If 
the  working  load  be  made  ^th  of  the  breaking  load,  as  in  the 
case  of  the  beam,  then  the  breaking  load  should  be  81,360  Ibs., 
and  the  strength  of  the  connecting-rod  should  be  such  that  it 
would  just  break  with  that  load  on  the  piston.  Now  the  tensile 
strength  of  the  weakest  cast-iron  is  about  12,000  Ibs.  per  square 
inch  of  section,  while  its  crushing  strength  is  about  five  times 
that  amount.  Dividing  81,361  Ibs.,  the  total  tensile  strength  of 
the  rod,  by  12,000,  the  tensile  strength  of  one  square  inch,  we 
get  about  V  square  inches  as  the  proper  area  of  the  smallest  part 
of  the  connecting-rod  when  of  cast-iron.  But  ^th  of  452  (which 
is  the  area  of  the  cylinder  in  square  inches)  is  13  square  inches, 
from  all  of  which  it  follows  that  while  the  main  beams  of  en- 
gines are  commonly  made  too  weak,  the  cast-iron  connecting- 
rods  are  commonly  made  too  strong.  This,  however,  is  partly 
done  for  the  purpose  of  balancing  the  weight  of  the  piston  and 
its  connections. 

FLY-WHEEL  SHAFT. 

The  fly-wheel  shaft  of  land  engines  is  usually  made  of  cast- 
iron.  The  following  is  the  rule  on  which  such  shafts  are  usually 
proportioned : — 


PROPEK   DIAMETER   OP    FLY-WHEEL    SHAFT.  239 

TO   FIND   THE   DIAMETER   OF   THE   FLY-WHEEL   SHAFT   AT   SMALLEST 
PAET,    WHEN   IT   IS   OF   CAST-IRON. 

RULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  by  the  length  of  the  crank  in  inches;  extract  the  cube 
root  of  the  product ;  finally  multiply  the  result  l>y  '3025. 
The  product  is  the  diameter  of  the  fly-wheel  shaft  at  the 
smallest  part  in  inches. 
Example  1. — "What  is  the  proper  diameter  of  the  fly-wheel 

shaft,  when  of  cast-iron,  in  the  case  of  an  engine  with  a  diameter 

of  cylinder  of  64  inches  and  a  stroke  of  8  feet  ? 

64  —  diameter  of  the  cylinder  in  inches 
64 

4096  =  square  of  the  diameter 
48  =  length  of  crank  in  inches 


196608 

68-15  =  ^196608  and    58-15  x  -3025  =  17-59,  which  is  the  proper 
diameter  of  the  fly-wheel  shaft  at  the  smallest  part. 

Example  2. — What  is  the  proper  diameter,  at  the  smallest 
part,  of  the  cast-iron  fly-wheel  shaft  of  an  engine,  with  a  diameter 
of  cylinder  of  40  inches,  and  5  feet  stroke  I 

40  =  diameter  of  cylinder  hi  inches 
40 

1600  =  square  of  diameter  of  cylinder 
80  =  length  of  crank  hi  niches 


48000 

86-30  =  ^48000  and  36-30  x  -3025  =  10-98,  which  is  the  proper  diam- 
eter of  the  shaft  in  niches. 

ME.  WATT'S  BULE  FOE  THE  NECKS  OF  HIS  CEANK  SHAFTS. 
RULE. — Multiply  the  area  of  the  piston  in  square  inches  1y  the 
pressure  on  each  square  inch  (and  which  Mr.  Watt  took  at 
12  ZJs.),  and  ~by  the  length  of  the  crank  in  feet.  Divide  the 
product  ~by  31-4,  and  extract  the  cube  root  of  the  quotient, 
which  is  the  proper  diameter  of  the  shaft  in  inches. 


240  PROPORTIONS    OF   STEAM-ENGINES. 

Example  1.— What  is  the  proper  diameter  of  the  fly-wheel 
shaft  in  an  engine,  with  a  cylinder  64  inches  diameter  and  8  feet 
stroke,  the  pressure  on  the  piston  being  taken  at  12  Ibs.  per 
square  inch  ? 

The  area  of  a  cylinder  64  inches  diameter  is  3,217  square 
inches,  which  multiplied  by  12  =  38,604,  and  this  multiplied  by 
4,  which  is  the  length  of  the  crank  in  feet,  is  154,416.  This 
divided  by  31'4  —  4,917'7,  the  cube  root  of  which  is  17'01 
inches. 

Example  2. — "What  is  the  right  diameter,  according  to  Mr. 
"Watt's  rule,  of  the  fly-wheel  shaft  of  an  engine,  with  a  24-inch 
cylinder,  5  feet  stroke,  and  with  a  pressure  of  12  Ibs.  on  each 
square  inch  of  the  piston  ? 

The  area  of  the  cylinder  is  452  square  inches,  which  multi- 
plied by  12  =  5,424,  and  this  multiplied  by  2£,  which  is  the 
length  of  the  crank  in  feet  —  13,560,  which  divided  by  31'4  = 
431,  the  cube  root  of  which  is  7£  inches,  which  is  the  proper 
diameter  of  the  shaft.  In  Mr.  Caird's  engine  the  diameter  is  8 
inches. 


TO  FIND  THE  PEOPEB  THICKNESS  OF  THE  LAEGE  EYE  OF  THE 
CBANK  FOE  FLY-WHEEL  SHAFT,  WHEN  OF  OAST-IEON. 

RULE. — Multiply  the  square  of  the  length  of  the  crank  in  inches 
ly  1'561,  and  then  multiply  the  square  of  the  diameter  of  the 
cylinder  in  inches  l>y  '1235;  multiply  the  sum  of  these  prod- 
ucts l>y  the  square  of  the  diameter  of  the  cylinder  in  inches  ; 
divide  this  product  2>y  666*283  /  divide  this  quotient  by  the 
length  of  the  crank  in  inches  ;  finally  extract  the  cube  root  of 
the  quotient.  The  result  is  the  proper  thickness  of  the  large 
eye  of  crank  for  fly-wheel  shaft  in  inches,  when  of  cast-iron. 

Example  1. — Required  the  proper  thickness  of  the  large  eye 
of  crank  for  fly-wheel  shaft,  when  of  cast-iron,  of  an  engine 
whose  length  of  stroke  is  8  feet,  and  diameter  of  cylinder  64 
inches. 


THICKNESS  OF  LARGE  EYE  OF  CRANK.       241 

48  =  length  of  crank  in  inches 
48 

2304  =  square  of  length  of  crank  in  inches 
1-561  =  constant  multiplier 


8596-5 


64  =  diameter  of  cylinder  in  inches 
64 

4096  —  -f  S(luare  °f    diameter  of  cylinder 
(  in  inches 

•1235  =  constant  multiplier 


505-8 
3596-5 


4102-3  =  sum  of  products 
) 


409(5  _  j  square  of  the  diameter  of  the  cy- 
(  Under  in  niches 


=  666-283)16803020-8 


Length  of  crank  =  48)25219-045 
525-397 


and  ^525-397  =  8-07  which  is  the  proper  thickness  of  the  large  eye  of 
the  crank  in  inches,  when  of  cast-iron. 

Example  2. — Required  the  proper  thickness  of  the  large  eye 
of  the  crank  for  fly-wheel  shaft,  when  of  cast-iron,  of  an  en- 
gine, whose  length  of  stroke  is  5  feet,  and  diameter  of  cylinder 
40  inches. 

30  =  length  of  crank  in  inches 
30 

900  =  square  of  length  of  crank  in  inches 
1-561  =  constant  multiplier 

1404-9 
11 


242  PROPORTIONS   OF   STEA3I-ENGIKES. 

40  =  diameter  of  cylinder  in  inches 
40 

1600 

•1235  =  constant  multiplier 


197-6 
1404-9 

1602-5  =  sum  of  products 

1600    =  square  of  diameter  of  cylinder 

I  =  666-283)2564000-0 
Length  of  crank  —  30  inches  3848-2 

128-3 

and  ^/128'3  =  5  -04  inches  is  the  proper  thickness  in  this  engine  of  the 
large  eye  of  the  crank,  when  of  cast-iron. 

TO  FIND  THE  PROPER  BREADTH  OF  THE  WEB  OF  THE  CRANK  AT 
THE  CENTRE  OF  THE  FLY-WHEEL  SHAFT,  WHEN  OF  CAST-IRON, 
SUPPOSING  THE  BREADTH  TO  BE  CONTINUED  TO  THE  CENTRE 
'  OF  THE  SHAFT. 

RULE. — Multiply  the  square  of  the  length  of  the  crank  in  inches 
by  1-561,  and  then  multiply  the  square  of  the  diameter  of  the 
cylinder  in  inches  by  "1235  /  multiply  the  square  root  of  the 
sum  of  these  products  by  the  square  of  the  diameter  of  the 
cylinder  in  inches  ;  divide  the  product  by  23*04,  and  finally 
extract  the  cube  root  of  the  quotient.  The  final  result  is  the 
breadth  of  the  crank  at  the  centre  of  the  fiy-wheel  shaft, 
when  the  crank  is  of  cast-iron. 
Example  1. — What  is  the  proper  breadth  of  the  web  of  the 

crank  at  the  centre  of  fly-wheel  shaft,  when  of  cast-iron,  in  the 

case  of  an  engine,  with  a  diameter  of  cylinder  of  64  inches,  and 

length  of  stroke  8  feet  ? 

48  =  length  of  crank  in  inches 
48 

2304  =  square  of  length  of  crank 
1*561  =  constant  multiplier 

3596-5 


PROPER   BREADTH   OF   WEB    OP   CRANK.  243 

&l  =.  diameter  of  cylinder  in  inches 
64 

4096  =  square  of  diameter  of  cylinder 
•1235  —  constant  multiplier 

505-8 
3596-5 


4102-3  =  sum  of  products 


v/4102-3= 64-05  nearly 

4096  =  square  of  diameter  of  cylinder  in  inches. 

23-04)262348-80(11395-34 
2304 


3214 
2304 

9108 
6912 

21968 
20736 

12320 
11520 


8000 
6912 

10880 
9216 

1664 

•$11395-34  =  22-5  inches,  which  is  the  proper  breadth  of  the  web  of  the 
crank,  when  of  cast-iron,  supposing  the  breadth  to  be  continued  to  the 
centre  of  the  fly-wheel  shaft. 

Example  2.— "What  is  the  proper  breadth  of  the  web  of  a 
cast-iron  crank  at  the  centre  of  the  fly-wheel  shaft  (supposing  it 
to  be  so  far  extended),  in  the  case  of  an  engine  with  40  inches 
diameter  of  cylinder  and  5  feet  stroke  ? 


244  PROPORTIONS    OF    STEAM-ENGINES. 

80  —  length  of  crank  in  inches 
30 

900  =  square  of  length  of  crank  in  inches 
1-561  =  constant  multiplier 


1404-9 

40  =  diameter  of  cylinder  in  inches 
40 

1600  =  square  of  diameter  of  cylinder 
•1235  =  constant  multiplier 

197-6 

1602-5  =  sum  of  products 
v/1602-5  =  40-3  nearly 
1600 


23-04)64480-0 


2798-6  nearly 

•$'2798-6  =  14-09,  which  is  the  proper  breadth  in  inches  of  a  cast  iron 
crank  in  an  engine  of  this  size,  supposing  the  breadth  to  be  continued 
to  the  fly-wheel  shaft. 


TO   FIND   THE    PEOPEK  THICKNESS    OP    THE   WEB    OF    A    OAST-IKON 
CRANK   AT   THE   CENTEE   OF   THE   FLY-WHEEL  SHAFT. 

RULE. — Multiply  the  square  of  the  length  of  the  crank  in  inches 
T>y  1'561,  and  then  multiply  the  square  of  the  diameter  of  the 
cylinder  in  inches  l>y  '1235 ;  multiply  the  square  root  of  the 
sum  of  these  products  ly  the  square  of  the  diameter  of  the 
cylinder  in  inches;  divide  the  product  'by  1'32;  finally 
extract  the  cube  root  of  the  quotient.  The  result  is  the  proper 
thicTcness  of  the  web  of  a  cast-iron  crank  in  inches  at  the  cen- 
tre of  the  fly-wheel  shaft,  supposing  the  thickness  to  ~be  ex- 
tended to  that  point. 

Example  1. — Required  the  proper  thickness  of  the  web  of  a 
cast-iron  crank  at  the  centre  of  the  fly-wheel  shaft  (supposing  it 


PROPER  THICKNESS  OF  WEB  OF  CRANK.      245 

to  be  so  far  extended),  in  the  case  of  an  engine  with  64  inches 
diameter  of  cylinder,  and  8  feet  stroke. 

48  =  length  of  crank  in  inches 
48 

2304  =  square  of  the  length  of  crank 
1-561  =  constant  multiplier 


3596-5 

64  =:  diameter  of  cylinder  in  inches 
64 

4096  =r  square  of  diameter  of  cylinder 
•1235  =:  constant  multiplier 


505-8 
3596-5 

4102-3  =  sum  of  products 
and  ^/4102-3  =  64-05  nearly 

4096  =  square  of  diameter  of  cylinder 


1422-33 
and  v'  1423-33  =  11-25 

Example  2.  —  "What  is  the  proper  thickness  of  the  web  of  a 
cast-iron  crank  at  centre  of  fly-wheel  shaft  (supposing  it  to  be  so 
far  extended),  in  the  case  of  an  engine  with  40  inches  diameter 
of  cylinder,  and  5  feet  stroke? 

30  =  length  of  crank  in  inches 
30 


_  j 
-  \ 


square    of   length    of   crank    in 
inches 


1-561  —  constant  multiplier 
1404-9 


246  PROPORTIONS   OF   STEAM-ENGINES. 

40  =  diameter  of  cylinder  in  inches 
40 

1600  =  square  of  diameter  of  cylinder 
•1235  =  constant  multiplier 


197-6 
1404-9 


1602-5 

V 1602-5    =  40-3  nearly 
1600 


349-8 

and  ^349-8  =  7'04,  which  is  the  proper  thickness  in  inches  of  the  web 
of  a  cast-iron  crank  for  this  engine,  measuring  at  the  centre  of  the  fly- 
wheel shaft. 

CRANK  PIN. 

The  crank  pins  of  land  engines  having  cast-iron  cranks,  are 
generally  made  of  cast-iron,  and  are  in  diameter  about  one-sixth 
of  the  diameter  of  the  cylinder. 

MILL  GEARING. 

Boulton  aad  Watt,  by  whom  the  present  system  of  iron 
gearing  was  introduced,  proportioned  their  wheels  on  the  follow- 
ing consideration : — '  That  a  bar  of  cast-iron  1  inch  square  and 
12  inches  long,  bears  600  Ibs.  before  it  breaks ;  1  inch  long  will 
bear  7,200  Ibs.,  and  TVth  of  this  =  480  Ibs.,  is  the  load  which 
should  be  put  on  the  wheel,'  for  each  square  inch  in  section  of 
the  tooth.  Boulton  and  Watt's  rule  for  the  strength  of  geared 
wheels  is  consequently  as  follows : — If  H  =  the  actual  horses' 
power  which  the  wheel  has  to  transmit ;  d,  the  diameter  of  the 
wheel  in  feet,  and  r,  the  revolutions  of  the  wheel  per  minute ; 
then  H  x  306 

— -5— =  the  strength,  and  the  strength  divided  by  the 

breadth  in  inches  =p\  or  the  square  of  the  pitch  in  inches. 


PROPER   PROPORTIONS   OF   TOOTHED   WHEELS.        247 


Hence  H  =?  x  *  X  d  X  rand  p  =4/H  X  3°6  ,  which  equations 
306  V  5  x  d  x  r' 

put  into  words  are  as  follows : 

TO  FIND  THE  NUMBER  OF  ACTUAL  HOESES  POWER  WHICH  A  GIVEN 
WHEEL  WILL  TRANSMIT,  ACCORDING  TO  BOULTON  AND  WATT'S 
PRACTICE. 

RULE. — Multiply  the  square  of  the  pitch  in  inches  ~by  the  "breadth 
of  the  wheel  in  inches,  by  its  diameter  in  feet,  and  by  tlie 
number  of  revolutions  it  makes  per  minute,  and  divide  the 
product  ~by  the  constant  number  306.  The  quotient  is  the 
number  of  actual  horses*  power  which  the  wheel  will  safely 
transmit,  according  to  Boulton  and  Waffs  practice. 

TO  FIND  THE  PROPER  PITCH  OF  A  WHEEL  IN  INCHES  TO  TRANS- 
MIT A  GIVEN  POWER,  ACCORDING  TO  BOULTON  AND  WATT'S 
PRACTICE. 

EULE.— Multiply  the  breadth  of  the  teeth  in  inches  by  the  diam- 
eter of  the  wheel  in  feet,  and  by  the  number  of  revolutions  it 
makes  per  minute,  and  reserve  the  product  as  a  divisor.  Next 
multiply  the  number  of  actual  horses'  power  which,  the  wheel 
has  to  transmit  by  the  constant  number  306,  and  divide  the 
product  by  the  divisor  found  as  above.  Finally,  extract  the 
square  root  of  the  quotient,  which  is  the  proper  pitch  of  the 
wheel  in  inches,  according  to  Boulton  and  Wattfs  practice. 

Instead,  however,  of  reckoning  the  strain  in  horses'  power,  it 
is  preferable  to  reckon  it  as  a  pressure  or  weight  applied  to  the 
acting  tooth  of  the  driving  wheel.  If  t  =  the  thickness  of  the 
tooth  in  inches,  w  =  the  pressure  upon  it  hi  Ibs.,  and  c  a  con- 
stant multiplier,  which  for  cast-iron  is  '025,  for  brass,  '035,  and 
for  hard  wood,  -038,  then  t  =  c  ^Jw,  by  which  formula  we  can 
easily  find  the  proper  thickness  of  the  tooth,  and  twice  the 
thickness  of  the  tooth  with  the  proper  allowance  for  clearance, 
gives  the  pitch.  This  formula  put  into  words  is  as  follows : — 


248  PROPORTIONS   OF   STEAM-ENGINES. 

TO  FIND  THE  PROPER  THICKNESS  OF  TOOTH  OF  A  CAST-IRON 
WHEEL  TO  TRANSMIT  WITH  SAFETY  AST  GIVEX  PRESSURE. 

EULE. — Multiply  the  square  root  of  the  pressure  in  pounds  act- 
ing at  the  pitch  line  T)y  the  constant  number  '025.  The 
product  is  the  proper  thickness  of  the  tooth  in  inches. 

Example  1. — What  is  the  proper  thickness  of  the  teeth  of  a 
cast-iron  wheel  moved  by  a  pressure  of  233*33  Ibs.  at  the  pitch 
circle  ? 

Here  V  233-33  =  15-27,  and  this  multiplied  by  '025  =  -381, 
which  is  the  proper  thickness  of  the  teeth  in  inches. 

Example  2.— What  is  the  proper  thickness  of  the  teeth  of  a 
cast-iron  wheel  which  is  moved  round  by  a  pressure  of  46,666'6 
Ibs.  at  the  pitch  circle  ? 

It  will  be  easiest  to  solve  this  question  by  means  of  logarithms. 
As  the  index  of  the  logarithm  is  always  one  less  than  the  number 
of  places  above  unity  filled  by  the  number  of  which  the  logarithm 
has  to  be  found ;  and  as  there  are  five  such  places  in  the  number 
46,666-6,  it  follows  that  the  index  of  the  logarithm  will  be  4,  and 
the  rest  of  the  logarithm  will  be  found  by  looking  for  the  nearest 
number  to  46,666-6  in  the  tables,  and  which  number  will  be 
4,666,  the  logarithm  answering  to  which  is  668945.  The  residue 
6-6,  however,  has  not  yet  been  taken  into  account,  and  to  include 
it  we  must  multiply  the  number  found  opposite  to  the  logarithm 
in  the  column  marked  D,  commonly  introduced  in  logarithmic 
tables  (and  which  is  a  column  of  common  differences),  by  the 
number  we  have  not  yet  reckoned,  namely,  6-6 ;  and  cut  off  a 
number  of  figures  from  the  product  equal  to  those  in  the  mul- 
tiplier, adding  the  residue  to  the  logarithm,  which  will  thereupon 
become  the  correct  logarithm  of  the  whole  quantity.  The  com- 
mon difference  in  this  case  is  93,  which  multiplied  by  6-6  gives 
613'8,  and  cutting  off  the  3*8  we  add  the  61  to  the  logarithm 
already  found,  which  then  becomes  4-669006.  Dividing  this  by 
2,  we  get  2-334503,  which  will  be  the  logarithm  of  the  number 
that  is"  the  square  root  of  46,666-6.  As  the  index  of  the  loga- 
rithm is  2,  there  will  be  three  places  above  unity  in  the  number, 
and  looking  now  in  the  logarithm  tables  for  the  number  answer- 


PROPER   PROPORTIONS    OF   TOOTHED    WHEELS          249 

ing  to  the  logarithm  nearest  334503,  we  get  the  number  216,  the 
logarithm  of  which  is  334454.  The  number  216  is  consequently 
the  square  root  of  46,666'6  very  nearly,  as  to  extract  the  square 
root  by  logarithms,  we  have  only  to  divide  the  logarithm  of  the 
number  by  2,  and  the  number  answering  to  the  new  logarithm 
thus  found  will  be  the  square  root  of  the  original  number.  Now 
216  multiplied  by  '025  =  5'400,  which  consequently  is  the  thick- 
ness in  inches  of  each  of  the  teeth  of  this  wheel. 


GENERAL   RULES   REGARDING   GEARING. 

The  pitch  should  be  in  all  cases  as  fine  as  is  consistent  with 
the  required  strength.  "When  the  velocity  of  the  motion  exceeds 
3£  feet  per  second,  the  larger  of  the  two  wheels  should  be  fitted 
with  wooden  teeth,  the  thickness  of  which  should  be  a  little 
greater  than  that  of  the  iron  teeth.  The  breadth  of  the  teeth  in 
the  direction  of  the  axis  varies  very  much  in  practice.  But 
where  the  velocity  does  not  exceed  5  feet  per  second,  a  breadth 
of  tooth  in  the  line  of  the  axis  equal  to  four  times  the  thickness 
of  the  tooth  will  suffice.  This  is  nearly  the  same  thing  as  a 
breadth  equal  to  twice  the  pitch.  Where  the  velocity  at  the 
pitch  circle  is  greater  than  5  feet  per  second,  the  breadth  of  the 
teeth  should  be  5  tunes  the  thickness  of  tooth,  the  surfaces  being 
kept  well  greased.  But  if  the  teeth  be  constantly  wet,  the 
breadth  should  be  6  times  the  thickness  of  tooth  at  all  velocities. 
The  best  length  of  the  teeth  is  fths  of  the  pitch,  and  the  length 
should  not  exceed  fths  of  the  pitch,  and  the  effective  breadth 
of  the  teeth  should  not  be  reckoned  as  exceeding  twice  the 
length ;  any  additional  breadth  being  good  for  wear,  but  not  for 
strength.  In  the  Soho  practice  the  length  of  the  teeth  is  made 
-j^ths  of  the  pitch  outside,  and  ^ihs  of  the  pitch  inside  of  the 
pitch  circle,  the  whole  length  being  -j^ths  or  fths  of  the  pitch. 
The  London  practice  is  to  divide  the  pitch  into  12  parts,  and  to 
adjust  the  length  of  the  tooth  by  allowing  ^ths  without,  and 
within  the  pitch  circle,  the  entire  length  of  tooth  being 
of  the  pitch.  The  projection  of  the  teeth  beyond  the  pitch 
circle  w.ill  be  |th  of  the  pitch,  and  the  surface  in  contact  between 
11* 


250  PROPORTIONS    OF   STEAM-ENGINES. 

the  teeth  of  the  two  wheels  will  be  half  the  pitch.  About  $th 
of  the  pitch  should  be  left  unoccupied  at  the  bottom  of  the  teeth 
for  clearance. 

"With  regard  to  the  least  number  of  teeth  that  is  admissible  in 
the  smaller  of  two  wheels  working  together,  12  to  18  teeth  will 
answer  well  enough  in  crane  work,  where  a  pinion  is  employed 
to  give  motion  to  a  AvheeJ  at  a  low  rate  of  speed.  But  for  quick 
motions,  a  pinion  driven  by  a  wheel  should  never  have  less  than 
from  30  to  40  teeth. 

The  best  form  of  teeth  is  the  epicycloidal,  and  in  general  the 
proper  curve  is  obtained  by  rolling  a  circle  of  wood  carrying  a 
pencil  on  another  circle  of  wood  answering  to  the  pitch  circle, 
the  point  of  the  tooth  being  described  by  the  rolling  circle  trav- 
ersing the  outside  of  the  pitch  line,  and  the  root  by  traversing 
the  inside  of  the  pitch  line.  The  diameter  of  the  rolling  circle 
should  be  2*22  times  the  pitch.  Some  teeth  are  not  epicycloi- 
dal, but  the  roots  are  radii  of  the  pitch  circle,  and  the  points 
are  described  with  compasses  from  the  pitch  centre  of  the  next 
tooth. 

In  the  following  table  will  be  found  the  thickness  and  pitch 
of  teeth  answering  to  different  amounts  of  load  or  pressure  at  the 
pitch  circles.  But  it  may  here  be  remarked  that  such  large 
pitches  as  12  and  13  inches  are  practically  not  used.  In  cases 
where  such  large  pressures  are  to  be  transmitted  as  answer  to 
pitches  over  5  inches  or  thereabout,  it  is  usual  to  distribute  the 
load  by  placing  two  or  more  parallel  wheels  upon  the  same  shaft, 
working  into  corresponding  pinions ;  and  it  is  also  usual  to  set 
the  teeth  of  each  wheel  a  little  in  advance  of  the  teeth  of  the 
wheel  next  it,  so  as  to  divide  the  pitch,  and  thus  render  the 
action  of  the  teeth  smoother  and  more  continuous. 


EXAMPLES   OP  HEAVY   GEARING. 


251 


PROPORTIONS   OF   THE   TEETH   OF   CAST-IRON   WHEELS. 


Pressure  in  Ibs. 
at  the 
pitch  circle. 

Pitch  of 
teeth  in  inches, 
allowing 
one-tenth  for 
clearance. 

Thickness 
of  teeth  in 
inches. 

Pressure  in  Ibs. 
at  the 
pitch  circle. 

Pitch  of 
teeth  in  inches, 
allowing 
one-tenth  for 
clearance. 

Thickness 
of  teeth  in 
inches. 

283-33 

•798 

•88 

11666-65 

5-6705 

2-7002 

849-95 

•981 

•467 

13999-98 

6-2118 

2-9580 

466-66 

1-134 

•540 

16883-31 

6-7099 

8-1952 

5S3-32 

1-268 

•604 

18666-64 

7-1728 

3-4156 

699-99 

1-388 

•661 

20999-97 

7-6079 

8-6228 

816-65 

1-5 

•716 

23383-3 

8-0194 

8-8188 

938-32 

1-604 

•763 

25666-63 

8-4109 

4-005? 

1049-98 

1-7 

•809 

27999-96 

8-7848 

4-1832 

1166-65 

1-793 

•854 

30838-29 

9-1470 

4-3557 

1283-31 

1-88 

•895 

82666-62 

9-4887 

4-5184 

1399-98 

1-964 

•935 

84999-95 

9-8218 

4-6770 

1516-64 

2-044 

.    -973 

87883-28 

10-1439 

4-8304 

1683-31 

2-121 

1-04 

89666-61 

10-4560 

4-9790 

1749-9T 

2-196 

1-045 

41999-94 

10-7592 

5-1284 

1866-64 

2-268 

1-08 

44838-27 

11-0540 

6-2638 

1983-3 

2-S88 

1-118 

46666-6 

11-8412 

5-4006 

2099-9T 

2-405 

1-145 

49999-98 

11-7381 

5-5896 

2216-63 

2-471 

1-177 

52383-26 

12-0103 

5-7192 

2333-3 

2-588 

1-208 

54666-59 

12-2749 

5-8452 

2449-96 

2-598 

1-287 

66999-92 

12-5341 

5-9686 

2566-63 

2-659 

1-266 

69833-25 

12-7883 

6-0897 

2683-29 

2-720 

1-295 

60666-58 

12-9310 

6-1576 

2799-96 

2-777 

1-822 

62999-91 

13-1773 

6-2749 

4666-66 

8-586 

1-7078 

65333-24 

18-8893 

6-8759 

6999-99 

4-8924 

2-0916 

67666-57 

13-6566 

6-5031 

9833-32 

5-0719 

2-4152 

69999-99 

13-WI1 

6-6143 

It  will  be  useful  to  illustrate  the  application  of  these  rules  to 
the  case  of  heavy  gearing  by  one  or  two  practical  examples. 

In  a  steamer  with  engines  by  Messrs.  Penn  and  Son  there 
are  two  cylinders  of  82£  inches  diameter  and  6  feet  stroke, 
giving  motion  to  a  toothed  wheel  14  feet  diameter  consisting  of 
four  similar  wheels  bolted  together,  the  teeth  being  12  inches 
broad  and  5'86  inches  pitch.  The  area  of  a  cylinder  82£  inches 
being  5,346  square  inches,  there  will  be  a  total  pressure  on  the 
piston — if  we  reckon  the  mean  average  pressure  upon  each  square 
inch  at  25  Ibs. — of  133,650  Ibs.  But  as  there  are  two  pistons,  the 
total  pressure  on  the  two  pistons  will  be  267,300  Ibs.  Now  the 
diameter  of  the  geared  wheel  being  14  feet,  its  circumference 
will  be  44  feet,  and  as  at  each  movement  of  the  pistons  up  and 
down  through  the  length  of  the  stroke,  or  through  a  distance 
of  12  feet,  the  wheel  makes  one  revolution,  or  moves  through  44 
feet,  the  pressure  at  the  circumference  of  the  wheel  will  be  lesa 


252  PROPORTIONS   OP    STEAM-ENGINES. 

than  that  on  the  pistons  in  the  proportion  in  which  44  exceeds 
12,  so  that  by  multiplying  267,300  by  12  and  dividing  the  product 
by  44  we  get  the  equivalent  or  balancing  pressure  at  the  circumfer- 
ence of  the  wheel,  and  which  is  69,073  Ibs.  As,  however,  this  load 
is  distributed  among  four  wheels,  there  will  only  be  one-fourth  of 
69,673,  or  17,418  Ibs.  to  be  borne  by  each  of  them.  According 
to  the  rule  we  have  given,  therefore,  the  square  root  of  17,418 
multiplied  by  '025  will  be  the  proper  thickness  of  each  tooth  in 
inches.  Now  ./1 7,41 8  — 132,  and  132  x  -025  =  3'3,  which  by  our 
rule  is  the  proper  thickness  of  the  tooth  in  inches,  and  twice  this, 
or  6'6,  with  one-tenth  or  '3  for  clearance,  will  be  the  pitch  =  6'9, 
whereas  the  actual  pitch  is  1  inch  less  than  this.  If  the  multi- 
plier be  made  '02,  instead  of  '025,  the  value  obtained  will  agree 
more  nearly  with  this  example,  as  132  x  '02  =  2'64,  which  will  be 
the  thickness  of  tooth,  and  2-64  x  2  =  5'28,  to  which  adding  ^th 
of  the  thickness  of  the  tooth  for  clearance,  or  '264,  we  get  5'544 
inches  as  the  pitch.  If  we  take  the  pressure  at  20  Ibs.  per  square 
inch  on  the  pistons  instead  of  25  Ibs.,  then  the  total  pressure  on 
the  two  pistons  will  be  213,840  Ibs.,  which  reduced  to  the  equiv- 
alent pressure  at  the  periphery  of  the  wheel  will  be  58,320  Ibs. 
The  fourth  of  this  is  14,580,  the  logarithm  of  which  is  4'163758, 
the  half  of  which  is  2-081879,  the  natural  number  answering  to 
which  is  120'7,  which  multiplied  by  '025  =  3'1175,  which  is  the 
proper  thickness  of  the  tooth  in  inches  for  this  amount  of  strain. 
It  will  be  seen,  therefore,  that  the  strength  which  our  rule  gives 
is  somewhat  greater  than  that  of  this  example. 

Let  us  now  take  an  example  by  a  different  maker,  and  we 
select  the  geared  engines  of  the  steamer  '  City  of  Glasgow,'  con- 
structed by  Messrs.  Tod  and  Macgregor.  There  were  two  cylin- 
ders in  this  vessel,  each  66  inches  diameter  and  5  feet  stroke,  and 
the  motion  was  communicated  from  the  crank  shaft  to  the  screw 
shaft  by  means  of  four  parallel  wheels,  V  feet  diameter,  8  inches 
broad,  and  4  inches  pitch.  The  area  of  a  cylinder  66  inches 
diameter  is  3,421  square  inches,  and  the  area  of  two  such  cylin- 
ders will,  consequently,  be  6,842  square  inches.  If  we  take  the 
pressure  urging  the  pistons  at  20  Ibs.  per  square  inch,  the  total 
pressure  on  the  pistons  will  be  136,840,  which  reduced  to  the 


EXAMPLES    OF   HEAVY   GEARING.  253 

pressure  at  the  periphery  of  the  wheel — which  moves  2'2  times 
faster  than  the  pistons — will  he  62,200  Ibs. ;  and  as  the  pressure 
is  divided  among  four  wheels  there  will  be  one-fourth  of  62,200, 
or  15,550  Ibs.  on  each.  The  logarithm  of  this  number  is  4-191430, 
the  half  of  which  is  2*095715,  the  natural  number  answering 
to  which  is  124-7,  and  124-7  multiplied  by  -025  =  3-1175,  which 
is  half  as  much  again  as  the  actual  strength  given  in  these 
wheels. 

We  may  take  still  another  example,  and  shall  select  the  case 
of  the  Tire  Queen,'  a  screw  yatch  constructed  by  Messrs. 
Eobert  Nap  ler  and  Sons.  In  this  vessel  there  are  two  cylinders, 
each  of  36  inches  diameter  and  36  inches  stroke,  and  the  motion 
is  communicated  from  the  crank  shaft  to  the  screw  shaft  through 
the  medium  of  three  parallel  wheels  8£  feet  diameter  placed  on 
the  end  of  the  crank  shaft.  The  pitch  of  the  teeth  is  3-55  inches, 
and  two  of  the  wheels  are  4  inches  broad,  and  one  of  them  6 
inches  broad.  The  two  narrow  wheels  may  be  reckoned  as  equiv- 
alent to  one  broad  one,  so  we  may  consider  the  strain  to  be 
divided  between  two  wheels.  The  area  of  each  cylinder  is 
1,018  square  inches,  and  if  we  reckon  two  cylinders  of  this  area, 
with  a  pressure  of  20  Ibs.  per  square  inch,  urging  the  piston  of 
each,  the  total  pressure  urging  the  pistons  will  be  40,720  Ibs. 
The  double  stroke  of  the  piston  is  6  feet,  and  the  circumference 
of  the  wheel  is  26g7  feet ;  and  as  the  wheel  revolves  once  while 
the  pistons  are  making  a  double  stroke,  the  relative  velocities  will 
be  6  and  26'7,  and  the  relative  pressures  26'7  and  6.  Multiply- 
ing, therefore,  40,720  by  6  and  dividing  by  26-7,  we  get  9,150  Ibs. 
as  the  pressure  at  the  circumference  of  the  wheel ;  and  as  this 
load  is  to  be  divided  between  two  wheels,  there  will  be  a  load 
of  4,575  Ibs.  upon  each.  The  logarithm  of  4,575  is  3-660391, 
the  half  of  which  is  1-830195,  the  natural  number  answering  to 
which  is  67-64,  which  multiplied  by  -025  gives  1-691  as  the  proper 
thickness  of  tooth  in  this  wheel.  Twice  1-691  is  3-382,  to  which 
if  weadd^th  of  the  thickness  of  the  tooth,  or  -169  for  clear- 
ance, we  get  3-55  as  the  proper  pitch  of  this  wheel,  and  this  is 
the  very  pitch  which  is  really  given.  In  this  case,  therefore 
the  rule  and  the  example  perfectly  correspond.  The  rule  gives 


254  PROPORTIONS   OP   STEAM-ENGINES. 

sufficient  strength  to  represent  the  mean  thickness  of  wooden 
and  iron  teeth — the  wooden  teeth  being  a  little  thicker,  and  the 
iron  teeth  a  little  thinner  than  the  amount  which  the  rule  pre- 
scribes. 


MARINE  ENGINES. 

The  rules  which  I  have  given  in  my  "Catechism  of  the 
Steam-Engine  "  for  fixing  the  proper  proportions  of  the  parts  of 
marine  engines,  take  into  account  the  pressure  of  the  steam  with 
which  the  engine  works.  But  in  order  that  the  proportions  thus 
arrived  at  may  be  more  easily  comparable  with  the  proportions 
subsisting  in  the  engines  of  different  constructors,  in  which 
the  pressure  is  assumed  as  tolerably  uniform,  it  will  be  more 
convenient  so  to  frame  the  rules  that  a  uniform  pressure  of  25 
Ibs.  per  square  inch  of  the  area  of  the  piston  shall  be  supposed  to 
be  at  all  times  existing.  In  cases  where  it  is  desired  to  ascer- 
tain the  dimensions  proper  for  a  greater  pressure  than  25  Ibs.,  it 
will  be  easy  to  arrive  at  the  right  result  by  taking  an  imaginary 
cylinder  of  as  much  greater  area  than  the  real  cylinder  as  the 
real  pressure  exceeds  the  assumed  pressure  of  25  Ibs.,  and  then 
by  computing  the  strengths  and  other  proportions  as  if  for  this 
imaginary  cylinder,  they  will  be  those  proper  for  the  real  cylin- 
der. Thus  if  it  be  desired  to  ascertain  the  strengths  proper  for 
an  engine  with  a  cylinder  of  30  inches  diameter,  and  with  a 
pressure  on  the  piston  of  100  Ibs.  on  the  square  inch,  the  end 
will  be  attained  if  we  determine  the  strengths  proper  for  an 
engine  of  60  inches  diameter,  and  with  25  Ibs.  pressure  on  the 
square  inch ;  for  the  area  of  the  larger  cylinder  being  four  times 
greater  than  that  of  the  smaller,  the  same  total  force  will  be  ex- 
erted with  one-fourth  of  the  pressure.  So,  in  like  manner,  if  it 
be  wished  to  ascertain  the  strengths  proper  for  an  engine  with  a 
cylinder  30  inches  diameter,  and  with  a  pressure  on  the  piston 
of  50  Ibs.  per  square  inch,  we  shall  find  them  by  determining 
the  proportions  suitable  for  an  engine  with  an  area  of  piston 
twice  greater  than  the  area  of  a  piston  30  inches  diameter,  and 
which  area  will  be  that  answering  to  a  diameter  of  42J  inches. 


DIMENSIONS   OP   THE   CROSSHEAD.  255 

By  this  mode  of  procedure  a  table  of  proportions  adapted  to  the 
ordinary  pressures  will  be  made  available  for  determining  the 
proportions  suitable  for  all  pressures,  as  we  have  only  to  fix  upon 
an  assumed  cylinder  which  shall  have  as  much  more  area  as  the 
intended  pressure  has  an  excess  of  pressure  over  25  Ibs.  per 
square  inch,  and  the  proportions  proper  for  this  assumed  cylin- 
der will  be  those  proper  for  the  real  cylinder  with  the  pressure 
intended.  In  this  way  the  strengths  fixed  for  marine  engines 
may  also  be  made  applicable  to  locomotives  and  to  high  and  low 
pressure  engines  of  every  kind.  In  the  following  rules,  there- 
fore, it  will  be  understood  the  strengths  and  other  proportions 
are  those  proper  to  an  assumed  pressure  on  the  piston,  including 
steam  and  vacuum,  of  25  Ibs.  per  square  inch,  and  the  computa- 
tions are  for  side  lever  engines,  but  for  the  most  part  are  appli- 
cable to  all  kinds  of  engines. 


CROSSHEAD. 

TO  FIND  THE  PROPER  THICKNESS  OF  THE  WEB  OF  THE  CROSS- 
HEAD  AT  THE  MIDDLE. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '072. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '072  =  2'880  inches,  which  is  the  proper 
thickness  of  the  web  of  the  crosshead  at  the  middle  in  this  en- 
gine. 

Example  2. — Let  64  inches  be  the  diameter  of  cylinder. 

Then  64  inches  x  '072  =  4' 608  inches,  which  is  the  proper 
thickness  of  the  web  of  the  crosshead  at  the  middle  in  this  en- 
gine. 

TO   FIND    THE    PROPER    THICKNESS    OF    THE   WEB     OF    THE   CROSS- 
HEAD   AT   THE   JOURNAL. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '061. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 
Then  40  inches  x  -061  =  2-440  inches,  which  is  the  proper 


256  PROPORTIONS   OF   STEAM-ENGINES. 

thickness  of  the  web  of  the  crosshead  at  the  journal  in  this  en- 
gine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x-061  =  3'904  inches,  which  is  the  proper 
thickness  of  the  web  of  the  crosshead  at  the  journal  in  this  en- 
gine. 

TO   FIND    THE    PROPER    DEPTH    OF   THE   WEB    OF   THE   CEOSSHEAD 
AT   THE   MIDDLE. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '268. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '268  =  10'720  inches,  which  is  the  proper 
depth  of  the  web  of  the  cTosshead  at  the  middle  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '268  =  l'T'152  inches,  which  is  the  proper 
depth  of  the  web  of  the  crosshead  at  the  middle  in  this  engine. 

TO  FIND  THE  PEOPEK  DEPTH  OF  THE  WEB  OF  THE  OEOSSHEAD 

AT  JOURNALS. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '101. 

Example  1. — Let  40  inches  be  the  diameter  of  cylinder. 

Then  40  inches  x  '101  =  4'040  inches,  which  is  the  proper 
depth  of  the  web  of  the  crosshead  at  journals  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '101  =  6'464  inches,  which  is  the  proper 
depth  of  the  web  of  the  crosshead  at  journals  in  this  engine. 

TO   FIND  THE  PROPER  DIAMETER   OF  THE  JOURNALS  OF  THE  CROSS- 
HEAD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  "086. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 
Then  40  inches  x  -086  =  3*440  inches,  which  is  the  proper 
diameter  of  the  journals  of  the  crosshead  in  this  engine. 
Example  2. — Let  64  inches  be  the  diameter  of  cylinder. 


DIMENSIONS   OF   THE   CROSSHEAD.  257 

Then  64  inches  x  '086  =  5'504  inches,  which  is  the  proper 
diameter  of  the  journal  of  crosshead  in  this  engine. 

TO  FIND  THE  PEOPER  LENGTH  OF  THE  JOURNALS  OF  THE  CROSS- 
HEAD. 

The  length  of  the  journals  of  the  crossheads  should  be  equal 
to  about  li  times  their  diameter,  but  on  the  whole  it  appears  to 
be  advisable  to  make  the  journals  of  the  crosshead  as  long  as 
they  can  be  conveniently  got. 

TO  FIND  THE  PEOPER  THICKNESS  OF  THE  EYE  OF  THE 
CROSSHEAD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '041. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '041  =  1'640  inches,  which  is  the  proper 
thickness  of  the  eye  of  the  crosshead  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '041  =  2'624  inches,  which  is  the  proper 
thickness  of  the  eye  of  the  crosshead  in  this  engine. 

TO  FIND  THE  PEOPER  DEPTH  OF  THE  EYE  OF  THE  CEOSSHEAD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '286. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '286  ==  11 -440  inches,  which  is  the  proper 
depth  of  the  eye  of  the  crosshead  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  -286  =  18-304  inches,  which  is  the  proper 
depth  of  the  eye  of  the  crosshead  in  this  engine. 

TO  FIND  THE  PBOPEB  DEPTH  OF  GIBS  AND  CUTTER  PASSING 
THROUGH  THE  OEOSSHEAD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '105. 


258  PROPORTIONS   OF   STEAM-ENGINES. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '105  =  4*200  inches,  which  is  the  proper 
depth  of  the  gibs  and  cutter  passing  through  the  crosshead  in 
this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '105  =  6'720  inches,  which  is  the  proper 
depth  of  the  gibs  and  cutter  passing  through  the  crosshead  in 
this  engine. 

TO   FIND   THE   PEOPEB  THICKNESS   OF   THE   GIBS   AND   CtTTTEE 
PASSING   THROUGH  THE   CEOSSHEAD. 

BULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  *021. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '021  —  '840  inches,  which  is  the  proper 
thickness  of  the  gibs  and  cutter  passing  through  the  crosshead 
in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '021  =  1/344  inches,  which  is  the  proper 
thickness  of  the  gibs  and  cutter  passing  through  the  crosshead 
in  this  engine. 


SIDE  RODS. 

TO   FIND   THE   PEOPEB  DIAMETER   OF   THE   CYLINDER   SIDE  RODS 
AT  THE  ENDS. 


.  —  Multiply  the  diameter  of  the  cylinder  in  inches  ly  '065. 

Example  1.  —  Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '065  =  2'600  inches,  which  is  the  proper 
diameter  of  cylinder  side  rods  at  ends  in  this  engine. 

Example  2.  —  Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  "065  =  4-160  inches,  which  is  the  proper 
diameter  of  the  cylinder  side  rods  at  ends  in  this  engine. 

The  diameter  of  the  side  rods  at  the  middle  should  be  about 


DIMENSIONS    OF   THE   SIDE   RODS.  259 

one-fourth  more  than  the  diameter  at  the  ends.  Thus  a  side  rod 
5  inches  diameter  at  the  ends  will  be  6J  inches  diameter  at  the 
middle. 

The  area  of  the  horizontal  section  of  iron  through  the  middle 
of  eye  of  side  rod  is  usually  about  one-half  greater  than  the  sec- 
tional area  of  the  side  rod  at  ends. 

TO   FIND   THE   PEOPEE   BEEADTH   OF   THE   BUTT   OF   THE   SIDE    EOD 
IN  INCHES. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '077. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '077  =  3'080  inches,  which  is  the  proper 
breadth  of  butt  of  side  rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '077  =  4'928  inches,  which  is  the  proper 
breadth  of  butt  in  this  engine. 

TO   FIND   THE   PEOPEE   THICKNESS   OF   THE  BUTT   OF   THE   SIDE 
EODS. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  "by  '061. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '061  =  2*440  inches,  which  is  the  proper 
thickness  of  the  butt  of  the  side  rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '061  =  3'904  inches,  which  is  the  proper 
thickness  of  the  butt  of  the  side  rod  in  this  engine. 

TO   FIND   THE   PEOPEB  MEAN   THICKNESS   OF   THE   8TEAP   OF   THE 
SIDE   EOD   AT   THE   CUTTEE. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  *032. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '032  =  1*280  inches,  which  is  the  proper 
mean  thickness  of  the  strap  of  side  rod  at  the  cutter  in  this 
engine. 


260  PROPORTIONS    OF   STEAM-ENGINES. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 
"  Then  64  inches  x  '032  =  2'048  inches,  which  is  the  proper 
mean  thickness  of  the  strap  of  side  rod  at  the  cutter  in  this 
engine. 

TO   FIND   THE   PROPER  MEAN  THICKNESS   OF   THE   STEAP   OF   SIDE 
ROD   BELOW   THE   CUTTER. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '023. 

Example  \. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '023 ='92  inches,  which  is  the  proper  mean 
thickness  of  the  strap  of  the  side  rod  below  the  cutter  in  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '023  =  1-472  inches,  which  is  the  proper 
mean  thickness  of  the  strap  of  the  side  rod  below  the  cutter  in 
this  engine. 

TO   FIND   THE   PROPER  DEPTH   OF   THE   GIB3   AND   CUTTER   OF 
SIDE   ROD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '08. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '08  =  3-20  inches,  which  is  the  proper 
depth  of  gibs  and  cutter  of  side  rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  -08  =  5'12  inches,  which  is  the  proper 
depth  of  gibs  and  cutter  of  side  rod  in  this  engine. 

TO   FIND   THE   PROPER  THICKNESS   OF    GIBS   AND   CUTTER   OF 
SIDE  ROD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  *016. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -016  =  -64  inches,  which  is  the  proper 
thickness  of  gibs  and  cutter  of  side  rod  in  this  engine. 

Example  2. — Let  64  inches  equal  the  diameter  of  cylinder. 

Then  64  inches  x  '016  =  1'02  inches,  which  is  the  proper 
thickness  of  gibs  and  cutter  of  side  rod  in  this  engine. 


DIMENSIONS   OF   THE   PISTON   ROD.  261 

PISTON  ROD. 
TO   FIND   THE   PROPER  DIAMETER   OF   THE   PISTON   ROD. 

RULE. — Divide  the  diameter  of  the  cylinder  in  inches  ~by  10. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  -s- 10  =  4'0  inches,  which  is  the  proper  diame- 
ter of  piston  rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  -r- 10  =  6-4  inches,  which  is  the  proper  diame- 
ter of  piston  rod  in  this  engine. 

TO   FIND   THE   PROPER   LENGTH   OF   THE   PART   OF   THE   PISTON   ROD 
IN  THE   PISTON. 

RULE. — Divide  the  diameter  of  the  cylinder  in  inches  ~by  5. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  -j-  5  =  8*0  inches,  which  is  the  proper  length 
of  the  part  of  the  piston  rod  in  the  piston  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  -j-  5  =  12'8  inches,  which  is  the  proper  length 
of  the  part  of  the  piston  rod  in  the  piston  in  this  engine. 

TO   FIND    THE    MAJOR    DIAMETER     OF     THE    PART    OF    THE   PISTON 
ROD   IN   THE   PISTON. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '14. 

Example  1. — Let  40  inches  equal  the  diameter  of  cylinder. 

Then  40  inches  x  '14  =  5'60  inches,  which  is  the  proper 
major  diameter  of  the  part  of  the  piston  rod  in  piston  in  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  -14  =  8-96  inches,  which  is  the  proper 
major  diameter  of  the  part  of  the  piston  rod  in  piston  in  this 
engine. 

TO    FIND    THE     MINOR    DIAMETER    OF     THE    PART    OF    THE    PISTON 
HOD   IN   THE   PISTON. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '115. 


262  PROPORTIONS   OF   STEAM-ENGINES. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -115  =  4*600  inches,  which  is  the  proper 
minor  diameter  of  the  part  of  the  piston  rod  in  piston  in  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  *115  =7'360  inches,  which  is  the  proper 
minor  diameter  of  the  part  of  the  piston  rod  in  piston  in  this 
engine. 

TO   FIND   THE    MAJOR    DIAMETER     OF    THE   PART     OF    THE    PISTON 
ROD   IN   THE   OROSSHEAD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '095. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '095  =  3'800  inches,  which  is  the  proper 
major  diameter  of  the  part  of  the  piston  rod  in  the  crosshead 
in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  *095  =  6'080  inches,  which  is  the  proper 
major  diameter  of  the  part  of  the  piston  rod  in  the  crosshead  in 
this  engine. 

TO     FIND    THE    MINOR    DIAMETER    OF     THE   PART    OF    THE   PISTON 
ROD   IN   CROSSHEAD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '09. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '09  =  3 '60  inches,  which  is  the  proper 
minor  diameter  of  the  part  of  the  piston  rod  in  crosshead  in  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '09  =  5'76  inches,  which  is  the  proper 
minor  diameter  of  the  part  of  the  piston  rod  in  crosshead  in  this 
engine. 

TO   FIND   THE   PROPER    DEPTH     OF    THE   CUTTER   THROUGH  PISTON. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  T>y  -085. 
Example  1.— Let  40  inches  be  the  diameter  of  the  cylinder. 


DIMENSIONS   OP   THE   CONNECTING-ROD.  263 

Then  40  inches  x  '085  =  3-400  inches,  which  is  the  proper 
depth  of  the  cutter  through  the  piston  in  this  engine. 

Example  2.— Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '085  =  5-440  inches,  which  is  the  proper 
depth  of  the  cutter  through  the  piston  in  this  engine. 

TO     FIND     THE    PEOPEE     THICKNESS    OF    THE    CUTTEE     THBOUGH 
PISTON. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '035. 

Example  1. — Let  40  inches  by  the  diameter  of  the  cylinder. 

Then  40  inches  x  '035  =  1-400  inches,  which  is  the  proper 
thickness  of  cutter  through  the  piston  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '035  =  2-240  inches,  which  is  the  proper 
thickness  of  cutter  through  piston  in  this  engine. 


CONNECTING-ROD. 

TO    FIND    THE    PEOPEE    DIAMETEE   OF    THE    CONNECTING-BOD    AT 
THE  ENDS. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  Try  -095. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -095  =  3-800  inches,  which  is  the  proper 
diameter  of  the  connecting-rod  at  the  ends  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  "095  =  6-080  inches,  which  is  the  proper 
diameter  of  the  connecting-rod  at  the  ends  in  this  engine. 

The  diameter  of  the  connecting-rod  at  the  middle  will  vary 
with  the  length,  but  is  usually  one-fifth  more  than  the  diameter 
at  the  ends.  Thus  a  connecting-rod  7*7  inches  diameter  at  the 
ends  will  be  9-25  inches  diameter  at  the  middle. 

TO    FIND    THE   MAJOE  DIAMETEB    OF    THE   PAET  OF  CONNECTING- 
BOD  IN  THE  OEOS8TAIL. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  -098. 


264  PROPORTIONS   OF   STEAM-ENGINES. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '098  =  3*920  inches,  which  is  the  proper 
major  diameter  of  the  part  of  the  connecting-rod  in  the  cross- 
tail  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '098  =  6'272  inches,  which  is  the  proper 
major  diameter  of  the  part  of  connecting-rod  entering  the  cross- 
tail  in  this  engine. 

TO   FIND   THE   PEOPEE    MINOR    DIAMETER     OF     THE    PAET   OF   CON- 
NECTING-ROD  ENTERING   THE   CEOS8TAIL. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '09. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '09  =  3 '60  inches,  which  is  the  proper 
minor  diameter  of  the  part  of  the  connecting-rod  in  the  cross- 
tail  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '09  =  5'76  inches,  which  is  the  proper 
minor  diameter  of  the  part  of  the  connecting  rod  in  the  cross- 
tail  in  this  engine. 

TO   FIND    THE    PROPER    BREADTH    OF    BUTT   OF    THE   CONNEOTtNG- 

EOD. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  -156. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -156  =  6*240  inches,  which  is  the  proper 
breadth  of  the  butt  of  connecting-rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '156  =  9'984  inches,  which  is  the  proper 
breadth  of  the  butt  of  the  connecting-rod  in  this  engine. 

TO     FIND    THE    PROPER    THICKNESS    OF    THE    BUTT    OF    THE   OON- 
NECTING-ROD. 

RULE. — Divide  the  diameter  of  the  cylinder  in  inches  "by  8. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  -4-  8  =  5-00  inches,  which  is  the  proper  thick- 
ness of  the  butt  of  the  connecting-rod  in  this  engine. 


DIMENSIONS    OF   THE   CONNECTING-ROD.  265 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 
Then  64  inches  -j-  8  =  8'00  inches,  which  is  the  proper  thick- 
ness of  the  butt  of  the  connecting-rod  in  this  engine. 

TO   FIND   THE   PEOPEE    MEAN    THICKNESS    OF   THE   STEAP   OF   CON- 
NECTING-BOD  AT   THE   CUTTEE. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '043. 

.Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '043  =  1'720  inches,  which  is  the  proper 
mean  thickness  of  the  connecting-rod  strap  at  the  cutter  in  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '043  =  2'752  inches,  which  is  the  proper 
mean  thickness  of  the  connecting-rod  strap  at  the  cutter  in  this 
engine. 

TO  FIND  THE  PEOPEE  MEAN  THICKNESS  OF  THE  CONNECTING-BOD 
STEAP   ABOVE   CUTTER. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '032. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -032  =  1-280  inches,  which  is  the  proper 
mean  thickness  of  the  connecting-rod  strap  above  the  cutter  in 
this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  "032  =  2'048  inches,  which  is  the  proper 
mean  thickness  of  the  connecting-rod  strap  above  the  cutter  in 
this  engine. 

TO  FIND  THE  PBOPEE  DISTANCE  OF  CTJTTEE  FEOM  END  OF  STRAP 
OF  CONNECTING-BOD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  Tyy  '048. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '048  =  1*920  inches,  which  is  the  proper 
distance  of  the  cutter  from  the  end  of  the  strap  of  the  connect- 
ing-rod in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 
12 


266  PPOPORTIONS    OF   STEAM-ENGINES. 

Then  64  inches  x  '048  =  3-072  inches,  which  is  the  proper 
distance  of  the  cutter  from  the  end  of  the  strap  of  the  connect- 
ing-rod in  this  engine. 

TO  FIND   THE   PEOPEB    DEPTH   OF   THE   GIBS   AND   CUTTER  PASSING 
THROUGH   THE   CEOSSTAIL. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  l>y  '105. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '105  =  4'20  inches,  which  is  the  proper 
depth  of  the  gibs  and  cutter  passing  through  the  crosstail  in  this 
engine. 

Example,  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '105  =  6'720  inches,  which  is  the  proper 
depth  of  the  gibs  and  cutter  passing  through  the  crosstail  in  this 
engine. 

The  thickness  of  the  cutters  passing  through  the  crosstail  will 
be  the  same  as  the  thickness  of  those  passing  through  the  cross- 
head. 

TO  FIND  THE  PBOPEB  DEPTH  OF  THE  GIBS  AND  OUTTEE  THEOUGH 
THE  BUTT  OF  THE  CONNEOTING-EOD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '11. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '11  =  4'40  inches,  which  is  the  proper 
depth  of  the  gibs  and  cutter  passing  through  the  butt  of  the 
connecting-rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '11  =  7'04  inches,  which  is  the  proper 
depth  of  the  gibs  and  cutter  passing  through  the  butt  of  the  con- 
necting-rod in  this  engine. 

TO   FIND   THE    THICKNESS     OF    THE    GIB8    AND    CUTTER    THEOUGH 
THE   BUTT   OF   THE    CONNECTING-BOD. 

RULE.— Multiply  the  diameter  of  the  cylinder  in  inches  by  -029. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 
Then  40  inches  x  "029  =  1*160  inches,  which  is  the  proper 


DIMENSIONS    OF   THE    SIDE   LEVER.  267 

thickness  of  the  gibs  and  cutter  passing  through  the  butt  of  the 
connecting-rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '029  =  1*856  inches,  which  is  the  proper 
thickness  of  the  gibs  and  cutter  passing  through  the  butt  of  the 
connecting-rod  in  this  engine. 

CROSSTAIL. 

The  crosstail  is  made  in  all  respects  the  same  as  the  cross- 
head,  except  that  the  end  journals,  where  the  crosstail  butts  fit 
on,  are  made  so  that  the  length  is  only  equal  to  the  diameter  of 
the  journal,  instead  of  being  about  !£•  times,  as  in  the  crosshead. 
But  as  the  crosstail  butts  do  not  work  on  these  journals  or  gudg- 
eons, but  are  keyed  fast  upon  them,  the  shorter  length  is  pre- 
ferable. The  butts  of  the  crosstail  have  the  eyes  nearly  twice 
the  diameter  of  the  journals,  or  more  accurately  1*8  times,  and 
the  butts  for  the  reception  of  the  straps  for  connecting  to  the 
side  lever  are  made  of  the  same  dimensions  as  the  butts  of  the 
side  rods. 

SIDE  LEVER  AND  STUDS  OR  CENTRES. 

The  side  lever  is  usually  made  of  cast-iron.  But  it  should  be 
in  all  cases  encircled  by  a  strong  wrought-iron  hoop,  thinned  at 
the  edge  so  that  it  may  be  riveted  or  bolted  all  along  to  a  flange 
cast  on  the  beam  for  this  purpose,  and  forming  an  extension  of 
the  usual  edge  bead.  The  proportions  given  in  the  rules  are 
those  of  the  common  cast-iron  side  levers  as  usually  constructed. 
But  the  strength  will  be  increased  three  times  if  wrought-iron 
be  substituted  for  cast  in  the  top  and  bottom  flanges  or  edge 
beads. 

TO  FIND  THE  PBOPEE  DEPTH  OF  THE  SIDE  LEVER  AOB088  THE 
CENTRE. 

ETTLE. — Multiply  the  length  of  the  side  lever  in  feet  ly  '7423  ; 
extract  the  cube  root  of  the  product  and  reserve  the  root  for  a 
multiplier.  Then  square  the  diameter  of  the  cylinder  in 


ZOO  PROPORTIONS    OF   STEAM-ENGINES. 

inches  ;  extract  the  cule  root  of  the  square.  The  product 
of  the  last  result,  and  the  reserved  multiplier,  is  the  depth  of 
the  side  lever  in  inches  across  the  centre. 

Example  1. — What  is  the  proper  depth  across  the  centre  of 
the  side  lever  in  the  case  of  an  engine  with  a  diameter  of  cylin- 
der of  64  inches  and  length  of  side  lever  of  20  feet  ? 

Here  20  =  length  of  side  lever  in  feet 
•7433  length  of  multiplier 


14-848  and  ^  14.846  =  2'458  nearly 

Also  64  =  diameter  of  cylinder 
64 

4096  and  ^  4096  =  16 

Hence  depth  at  centre  =  16  x  2-458  =  39'30  inches,  or  be- 
tween 39£  and  39  inches. 

Example  2. — What  is  the  proper  depth  across  the  centre  of 
the  side  lever  in  the  case  of  an  engine  with  a  diameter  of  cylin- 
der of  40  inches,  and  length  of  side  lever  of  15  feet. 

Here  15  —  length  of  side  lever 
•7423 


11-1345  and  ty  H'1345  =  2'232 

Also  40  =  diameter  of  cylinder 
40 

1600  and  -^  1600  =  11-69  which  x  2'232  =  26'09, 
or  a  little  over  26  inches. 

The  depth,  of  the  side  lever  at  the  ends  is  determined  hy  the 
depth  of  the  eyes  round  the  end  studs.  The  thickness  of  the 
side  lever  is  usually  made  about  -g^th  of  its  length,  and  the 
breadth  of  the  edge  bead  is  usually  made  about  -^  of  the  length 
of  the  lever  between  the  end  centres. 

TO   FIND   THE   PROPEE   DIAMETER   OF   THE   MAIN   CENTRE   JOURNAL. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  -183. 


DIMENSIONS    OF   THE    SIDE   LEVEE.  269 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '183  =  7'32  inches,  which  is  the  proper 
diameter  of  the  main  centre  journal  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  -183  =  11-712  inches,  which  is  the  proper 
diameter  of  the  main  centre  journal  in  this  engine. 

TO  FIND  THE  LENGTH  OF  THE  MAIN  CENTRE  JOURNAL. 

RULE. — Multiply  ike  diameter  of  the  cylinder  in  inches  ~by  '275. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -275  =  ll'OO  inches,  which  is  the  proper 
length  of  the  main  centre  journal  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '275  =  17'60  inches,  which  is  the  proper 
length  of  the  niain  centre  journal  in  this  engine. 

TO  FIND  THE  DIAMETEE  OF  THE  END  STUD9  OF  THE  SIDE  LETEE. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  /by  '07. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '07  =  2-80  inches,  which  is  the  proper 
diameter  of  the  end  studs  of  the  side  lever  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '07  =  4'48  inches,  which  is  the  proper 
diameter  of  the  end  studs  of  the  side  lever  in  this  engine. 

TO  FIND  THE  PEOPEE  LENGTH  OF  THE  END  STUDS  OF  THE  SIDE 
LEVEB. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  "076. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '076  =  8 '04  inches,  which  is  the  proper 
length  of  the  end  of  studs  of  the  side  lever  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  -076  =  4'86  inches,  which  is  the  proper 
length  of  the  end  studs  of  the  side  lever  in  this  engine. 

TO   FIND   THE   PEOPEE  DIAMETEE   OF   THE  AIB-PUMP   STUDS  IN  SIDE 
LEVEE. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  T>y  *045. 


270  PROPORTIONS   OF   STEAM-ENGINES. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  "045  =  1'80  inches,  which  is  the  proper 
diameter  of  the  stud  in  the  side  lever  for  working  the  air-pump 
of  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '045  =  2*88  inches,  which  is  the  proper 
diameter  of  the  air-pump  studs  in  the  side  levers  of  this  engine. 

TO   FIND  THE   PROPER   LENGTH  OF  THE  AIR-PUMP  STUDS  SET   IN   THE 
SIDE   LEVEE. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  Tiy  '049. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '049  =  1'96  inches,  which  is  the  proper 
length  of  the  air-pump  studs  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '049  —  3'136  inches,  which  is  the  proper 
length  of  the  air-pump  studs  in  this  engine. 

TO   FIND   THE   PROPEE  DEPTH   OF   THE   EYE   ROUND   THE   END  STUDS 
OF   SIDE   LEVEE. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  l>y  '074. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -OY4  =  2'96  inches,  which  is  the  proper 
depth  of  the  eye  round  the  end  studs  of  the  side  lever  in  this  en- 
gine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '074  =  4'736  inches,  which  is  the  proper 
depth  of  the  eye  round  the  end  studs  of  the  side  lever  in  this  en- 
gine. 

It  is  clear  that  the  diameter  of  the  end  stud  added  to  twice 
the  depth  of  the  metal  running  round  it  will  be  equal  to  the 
depth  of  the  side  lever  at  the  end 

Hence  2'1  +  twice  2'96  =  8'72  will  be  the  depth  in  inches  of 
the  side  lever  at  the  ends  in  the  engine  with  the  40-inch  cylin- 
der, and  4'48  +  twice  4'736  =  13-95  will  be  the  depth  in  inches 
of  the  side  lever  at  the  ends  in  the  engine  with  the  64-inch  cyl- 
inder. 


DIMENSIONS   OF  THE   CRANK.  271 

TO   FIND   THE   THICKNESS   OF   THE  EYE  BOUND   THE  END    STUDS   OF 
SIDE   LEVEE. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  *052. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '052  =  2'08  inches,  which  is  the  proper 
thickness  of  eye  of  side  lever  round  the  end  studs  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '052  =  3'328  inches,  which  is  the  proper 
thickness  of  eye  of  side  lever  round  the  end  studs  in  this  engine. 

THE  CRANK. 

TO   FIND  THE   PEOPEE  DIAMETEB   OF  THE   CBANK-PLN  JOUENAL8. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '142. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '142  =  5'680  inches,  which  is  tho  proper 
diameter  of  the  crank-pin  journal  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '142  =  9-080  inches,  which  is  the  proper 
diameter  of  the  crank-pin  journal  in  this  engine. 

TO  FIND  THE  PEOPEE  LENGTH   OF  THE  CEANK-PIN  JOUBNAI.. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '16. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '16  =  6*40  inches,  which  is  the  proper 
length  of  the  crank-phi  journal  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  *16  =  10'24  inches,  which  is  the  proper 
length  of  the  crank-pin  journal  in  this  engine. 

TO  FIND  THE  PEOPEE  THICKNESS  OF  THE  SMALL    EYE  OF  CBANK:. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '063. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 
Then  40  inches  x  '063  ==  2-52  inches,  which  is  the  proper 
thickness  of  the  small  eye  of  the  crank  in  this  engine. 


272  TKOPORTIONS    OF   STEAM-ENGINES. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 
Then  64  inches  x  '063  =  4'032  inches,  which  is  the  proper 
thickness  of  the  small  eye  of  the  crank  in  this  engine. 

TO  FIND  THE  PROPER  BREADTH  OF  THE  SMALL  EYE  OF  THE  CRANK. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '187. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  "187  =  7'48  inches,  which  is  the  proper 
breadth  of  the  small  eye  of  the  crank  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  *187  =  1T968  inches,  which  is  the  proper 
breadth  of  the  small  eye  of  the  crank  in  this  engine. 

TO     FIND    THE    PEOPER     THICKNESS   OF   THE   WEB   OF   CRANK,   SUP- 
POSING  IT   TO   BE   CONTINUED   TO   CENTRE   OF   CRANK   PIN. 

EIJLE. — Multiply  the  diameter  of  the  cylinder  in  inches  l>y  '11. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '11  =  4'40  inches,  which  is  the  proper 
thickness  of  the  web  of  crank  in  this  engine,  supposing  it  to  be 
continued  so  far  as  centre  of  pin. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '11  =  7'04  inches,  which  is  the  proper 
thickness  of  the  web  of  the  crank  in  this  engine,  supposing  that 
the  thickness  were  to  be  continued  to  the  centre  of  the  crank- 
pin  and  to  be  there  measured. 

TO  FIND  THE  PROPER  THICKNESS  OF  THE  WEB  OF  THE  CRANK, 
SUPPOSING  THE  THICKNESS  TO  BE  CONTINUED  TO  THE  CENTRE 
OF  THE  PADDLE-SHAFT. 

KULE. — Multiply  the  square  of  the  length  of  the  crank  in  inches 
ty  1'561,  and  then  multiply  the  square  of  the  diameter  of 
cylinder  in  inches  l>y  *1235.  Multiply  the  square  root  of  the 
sum  of  these  products  ly  the  square  of  the  diameter  of  the  cyl- 
inder in  inches  ;  divide  this  quotient  by  360  ;  finally  extract 
the  cube  root  of  the  quotient.  The  result  is  the  thickness  of 
the  web  of  the  crank  at  paddle  shaft  centre  in  inches. 
Example  1. — What  is  the  proper  thickness  of  tho  web  of  crank 


DIMENSIONS    OF   THE    CRANK.  273 

at  the  centre  of  the  paddle-shaft,  supposing  the  thickness  to  be 
continued  thither  and  there  measured,  in  the  case  of  an  engine 
with  a  diameter  of  cylinder  of  64  inches  and  stroke  of  8  feet. 

48  =:  length  of  craiik  in  inches 
48 

2304 

1-561  constant  multiplier 


3596-5 
505-8    product  of  642  and  -1236 


4102-3 

64  =  diameter  of  cylinder 
64 

4096 
•1235 


505-8 

and  V4102-3  =  64  05  nearly 

4096  =  square  of  diameter 


360)262348-5 


728-75 

And  ^'728  =  9  nearly,  which  is  the  proper  thickness  in  inches  of  the 
crank  of  this  engine  measured  at  the  centre  of  the  paddle  shaft. 

Example  2. — What  is  the  proper  thickness  of  the  weh  of  crank 
at  paddle-shaft  centre  in  the  case  of  an  engine  with  a  cylinder 
40  inches  in  diameter  and  stroke  of  6  feet? 

30  =  length  of  crank  in  inches 
30 

900  =  square  of  length  of  crank 
1-561  =  constant  multiplier 

1404-9 
12* 


274  PROPORTIONS   OF    STEAM-ENGINES. 

40  =;  diameter  of  cylinder  in  inches 
40 

1600  =  square  of  diameter  of  cylinder 
235  =  constant  multiplier 


197-9 
1404-9 

1602-8  and  -j/1602'8  =  40-03 
1600 


177-9 

And  ty  177'9  i=  5-62,  which  is  the  proper  thickness  in  inches  of  the  web 
of  the  crank,  supposing  the  web  to  be  continued  to  the  centre  of  the 
paddle  shaft. 

TO  FIND  THE  PROPER  BREADTH  OF  THE  WEB  OF  THE  CRANK  AT 
PIN-CENTRE,  SUPPOSING  IT  TO  BE  CONTINUED  TO  THE  CENTRE 
OF  THE  CRANK-PIN. 

EULE. — Multiply  the  diameter  of  the  cylinder  T)y  '16.  The  prod- 
uct is  the  proper  "breadth  of  the  web  of  the  crank,  supposing 
the  web  to  be  continued  to  the  plane  of  the  centre  of  the  crank- 
pin. 

Example  1. — Let  the  diameter  of  the  cylinder  be  40  inches. 
Then  40  inches  x  '16  =  6'4,  which  is  the  proper  breadth  in 

inches  of  the  web  of  the  crank  in  the  plane  of  the  centre  of  the 

crank-pin  in  this  engine. 

Example  2. — Let  the  diameter  of  the  cylinder  be  64  inches. 
Then  64  inches  x  *16  =  10'24  inches,  which  is  the  proper 

breadth  of  the  web  of  the  crank  at  the  crank-pin  end  in  this 

engine. 

TO    FIND    THE     PROPER     BREADTH    OF    THE     CRANK    AT  PADDLE- 
CENTRE. 

KULE. — Multiply  the  square  of  the  length  of  crank  in  inches  by 
1*561,  and  then  multiply  the  square  of  the  diameter  of  cyl- 
inder in  inches  ly  '1235  ;  multiply  the  square  root  of  the 
sum  of  these  products  by  the  square  of  the  diameter  of  the  cyl- 


DIMENSIONS   OF   THE   CRANK.  275 

inder  in  inches;  divide  the  product  ly  45.  Finally,  extract 
the  cute  root  of  the  quotient. 

Example  1. — What  is  the  proper  breadth  of  the  crank  at 
paddle-centre  in  the  case  of  an  engine  with  a  diameter  of  cylin- 
der of  64  inches  and  stroke  of  8  feet  ? 

48    length  of  crank  in  inches 
48 

2304 

1-561  constant  multiplier 

3596-5 
505-8 


4102-3 

64    diameter  of  cylinder 
64 

4086 

•1235  constant  multiplier 


505-8 

and    i/4102-S  =  64-05  nearly 
4096 


45)262348-5 


5829-97  and  ^5829  =  18  nearly,  which  is 

the  proper  breadth  in  inches  of  the  web  of  the  crank  at  the  shaft-centre 
in  this  engine. 

Example1},.—  What  is  the  proper  breadth  of  crank  at  paddle- 
centre  in  the  case  of  an  engine  with  a  diameter  of  cylinder  of  40 
inches  and  stroke  of  5  feet  ? 

30  =  length  of  crank  in  inches 
30 

900=  square  of  length  of  crank 
1-561 

1404-9 


276  PROPORTIONS   OF   STEAM-ENGINES. 

40  =  diameter  of  cylinder 
40 


1600  =  square  of  diameter  of  cylinder 
•1235 


197-6 
1404-9 


1602-5 

and  ^1602-5  =  40'03 
1600 


45)64048 


1466-7 
'and  -^1466-7  =  11-24 nearly. 

The  purpose  of  taking  the  breadth  and  thickness  of  the  web 
of  the  crank  at  the  shaft  and  pin-centres  is  to  obtain  fixed  points 
for  measurement.  For,  although  the  web  of  the  crank  does  not 
extend  either  to  the  centre  of  the  shaft  or  to  the  centre  of  the 
pin,  it  can  easily  be  drawn  in  as  if  extending  to  those  points, 
and  the  breadth  and  thickness  being  then  laid  down  at  those 
points  the  proper  amount  of  taper  in  the  web  of  the  crank  will 
be  obtained. 

TO    FIND    THE    PEOPEE    THICKNESS    OF    THE    LAEGE    EYE    OF   THE 
OEANK. 

KITLE. — Multiply  the  square  of  the  length  of  the  crank  in  inches 
~by  1-561,  then  multiply  the  square  of  the  diameter  of  the  cyl- 
inder in  inches  ~by  '1235  ;  multiply  the  sum  of  these  products 
J>y  the  square  of  the  cylinder  in  inches ;  divide  the  quotient 
"by  the  length  of  the  crank  in  inches;  afterwards  divide  the 
product  l>y  1828'28.  Finally,  extract  the  cube  root  of  the 
quotient.  The  result  is  the  proper  thicTcness  in  inches  of  the 
large  eye  of  crank. 

Example  1. — What  is  the  proper  thickness  of  large  eye  of  the 
the  crank  in  the  case  of  an  engine  with  a  diameter  of  cylinder 
of  64  inches  and  stroke  of  8  feet? 


DIMENSIONS   OF   THE   CRANK  277 

48  =  length  of  crank  in  inches 
48 

2304  =  square  of  length  of  crank 
1-561  =  constant  multiplier 


3596-5 
505-8  =  product  of  64*  and   1235 


4102-3 

64  =  diameter  of  cylinder 
64 

4096 

•1235  =  constant  multiplier 


505- 


4102-3 

4096  =  square  of  diameter 


48)16803020-8 
1828-28)350062-94 
191-4Y 


and  JJ/191-47  =  5'7Y  nearly,  whicn  is  the  proper  thickness  of  the  large 
eye  of  the  crank  in  inches. 

Example  2. — "What  is  the  proper  thickness  of  the  large  eye 
of  crank  in  the  case  of  an  engine  with  a  diameter  of  cylinder  of 
40  inches  and  with  a  stroke  of  5  feet  ? 

80  =  length  of  crank  in  inches 
30 

900  =  square  of  length  of  crank 
1-561  constant  multiplier 

1404-9 


278  PROPORTIONS   OF   STEAM-ENGINES. 

« 

40  =  diameter  of  cylinder 
40 

1600 

•1235  constant  multiplier 

197-6 
1404-9  add 


1602-5 

1600  =  square  of  diameter 


1828-28)2564000 


30)1402-41 

46-74 

and  %/  46-74  =  3-60,  which  is  the  proper  thickness  in  inches  of  the  large 
eye  of  the  crank  in  this  engine. 


TO   FIND   THE   PROPER  DIAMETER  OF  THE  PADDLE-SHAFT  JOURNAL. 

KULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  by  the  length  of  crank  in  inches  ;  extract  the  cube  root 
of  the  quotient.  Finally,  multiply  the  result  ~by  '242.  The 
final  product  is  the  diameter  of  the  paddle-shaft  journal  in 
inches. 

Example  1. — What  is  the  proper  diameter  of  the  paddle- 
shaft  journal  in  the  case  of  an  engine  with  a  diameter  of  cylin- 
der of  64  inches  and  stroke  of  8  feet  ? 

64  =  diameter  of  cylinder  in  inches 
64 

4096  square  of  diameter  of  cylinder 
48  =  length  of  crank  in  inches 


196608 
and  ^196608  =  58-148,  and  58'148  x  -242  =  14-07  inches. 

Example  2. — What  is  the  proper  diameter  of  the  paddle- 
shaft  journal  in  the  case  of  an  engine  with  a  diameter  of  cylinder 
of  40  inches  and  a  stroke  of  5  feet? 


DIMENSIONS    OF   THE   PADDLE-SHAFT.  279 

40  =  diameter  of  cylinder 

1600  =  square  of  diameter  of  cylinder 
30  =  length  of  crank  in  inches 


48000  and  ^48000  =  36'30 

and  36-30  x  -242  =  8'79  inches. 

TO  FETD  THE  PEOPEE  LENGTH  OF  THE  PADDLE-SHAFT  JOUBNAL. 

EULE. — Multiply  the  square  of  the  diameter  of  the  cylinder  in 
inches  by  the  length  of  the  crank  in  inches;  extract  the  cube 
root  of  quotient;  multiply  the  result  by  '303.  The  product 
is  the  length  of  the  paddle-shaft  journal  in  inches. 

Example  1. — "What  is  the  proper  length  of  the  paddle-shaft 
journal  in  the  case  of  an  engine  with  a  diameter  of  cylinder  64 
inches  and  stroke  8  feet? 

64  =  diameter  of  cylinder 
64 

4096  =  square  of  diameter  of  cylinder 
48  =  length  of  crank  in  inches 


196608     and  ^196608  =  58'148 
Length  of  journal  =  68-148  x  -303  =  17'60  inches. 

Example  2. — "What  is  the  proper  length  of  the  paddle-shaft 
journal  in  the  case  of  an  engine  with  a  diameter  of  cylinder  of 
40  feet  and  stroke  of  5  feet  ? 

40 
40 

1600 
30 


48000  and  V  48000=36-30  x  -303=10-99. 

It  will  be  seen  from  these  examples  that  the  length  of  the 
paddle-shaft  journals  is  1J  times  the  diameter.  The  paddle- 
shafts,  cranks,  and  all  the  other  working  parts  of  marine 


280  PROPORTIONS    OF   STEAM-ENGINES. 

engines  are  made  of  wrought-iron,  except  the  side  levers, 
which  are  of  cast-iron,  and  the  air-pump  rod,  which  is  of 
copper  or  brass. 

THE  AIR-PUMP. 

TO   FIND   THE   PEOPEE  DIAMETEE   OF    AIE-PUMP. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '6. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '6  =  24'0  inches,  which  is  the  proper  diam 
eter  of  the  air-pump  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '6  =  38-4  inches,  which  is  the  proper  diam- 
eter of  the  air-pump  in  this  engine. 

AIR-PUMP  ROD. 

TO   FIND   THE   PEOPEE  DIAMETEE  IN  INCHES   OF  THE  AIE-PUMP 
EOD   WHEN   OF   COPPER. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  T)y  '067. 

Example  1. — Let  the  diameter  of  the  cylinder  be  40  inches. 

Then  40  x  '067  =  2'68  inches,  which  is  the  proper  diameter 
of  the  air-pump  rod  when  of  copper  in  this  engine. 

Example  2. — Let  the  diameter  of  the  cylinder  be  64  inches. 

Then  64  x  '067  =  4'28  inches,  which  is  the  proper  diameter 
of  the  air-pump  rod  when  of  copper  in  this  engine. 

TO   FIND   THE   PEOPEB  DEPTH   OF   GLB8   AND   CUTTEE   THROUGH 
THE   AIE-PUMP  CEOSSHEAD  IN  INCHES. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  -063. 

Example  1. — Let  the  diameter  of  the  cylinder  be  40  inches. 

Then  40  x  '063  =  2'52  inches,  which  is  the  proper  depth  of 
gibs  and  cutter  through  the  air-pump  crosshead  in  this  engine. 

Example  2. — Let  the  diameter  of  the  cylinder  be  64  inches. 

Then  64  x  '063  =  4-03  inches,  which  is  the  proper  depth  of 
gibs  and  cutter  through  the  air-pump  crosshead  in  this  engine. 


DIMENSIONS    OF   PARTS    OP   THE   AIR-PUMP.  281 

TO   FIND   THE   PEOPEB  THICKNESS   OF   GIBS   AND    CtJTTEE 
THEOUGH  AIE-PUMP   OEOSSHEAD   IN   INCHES. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '013. 

Example  1. — Let  the  diameter  of  the  cylinder  be  40  inches. 

Then  40  x  '013  =  '52  inches,  which  is  the  proper  thickness 
of  gibs  and  cutter  through  the  air-pump  crosshead  in  this  engine. 

Example  2.— Let  the  diameter  of  the  cylinder  be  64  inches. 

Then  64  x  '013  =  '83  inches,  which  is  the  proper  thickness 
of  gibs  and  cutter  through  the  air-pump  crosshead  in  this  engine. 

TO   FIND   THE   PEOPEE  DEPTH   IN  INCHES   OF   THE   CUTTEE 
THEOUGH  THE    AIB-PUMP  BUCKET. 

KITLE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '051. 

Example,  1. — Let  the  diameter  of  the  cylinder  be  40  inches. 

Then  40  x  '061  =  2'04  inches,  which  is  the  proper  depth 
of  the  cutter  through  the  air-pump  bucket  in  this  engine. 

Example  2. — Let  the  diameter  of  the  cylinder  be  64  inches. 

Then  64  x  '051  =  3'26  inches,  which  is  the  proper  depth  of 
the  cutter  through  the  air-pump  bucket  in  this  engine. 

TO   FIND  THE   PEOPEE  THICKNESS   OF   THE   CUTTEE  THBOUGH  THE 
AIR-PUMP  BUCKET    IN   INCHES. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '021. 

Example  1. — Let  the  diameter  of  the  cylinder  be  40  inches. 

Then  40  x  "021  =  *84  inches,  which  is  the  proper  thickness 
of  the  cutter  through  the  air-pump  bucket  in  this  engine. 

Example  2. — Let  the  diameter  of  the  cylinder  be  64  inches. 

Then  64  x  *021  =  1'34  inches,  which  is  the  proper  thickness 
of  the  cutter  through  the  air-pump  bucket  in  this  engine. 

The  cutter  through  the  air-pump  bucket  should  be  always 
made  of  brass  or  copper,  but  the  gibs  and  cutter  through  the  air- 
pump  crosshead  will  be  of  iron.  The  air-pump  bucket  should 
always  be  of  brass,  and  it  is  advisable  to  insert  the  rod  into  the 
crosshead  and  also  into  the  bucket  with  a  good  deal  of  taper,  so 
as  to  facilitate  its  detachment  should  the  bucket  require  to  be 
taken  out.  It  is  usual  to  form  the  part  of  the  rod  projecting 


282  PROPORTIONS   OF   STEAM-ENGINES. 

through  the  crosshead  into  a  screw,  and  to  screw  a  nut  upon  it. 
This  also  is  a  common  practice  at  the  top  of  the  piston  rod  and 
at  the  bottom  of  the  connecting-rod. 

AIR-PUMP  CROSSHEAD. 

TO   FIND  THE   PROPER  DEPTH   OF   THE   EYE   OF  THE  AIR-PUMP 
OROSSHEAD. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '171. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '171  =  6'84  inches,  which  is  the  proper 
depth  of  eye  of  air-pump  crosshead  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  *171  =  10'944  inches  which  is  the  proper 
depth  of  the  eye  of  air-pump  crosshead  in  this  engine. 

TO  FIND  THE  PROPER  DEPTH  OF  THE  AIR-PUMP  OROSSHEAD  AT 
THE   MIDDLE   OF   THE    WEB. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ty  *161. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '161  =  6'44  inches,  which  is  the  proper 
depth  at  the  middle  of  the  web  of  the  air-pump  crosshead  hi  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  "161  =  10'30  inches,  which  is  the  proper 
depth  at  the  middle  of  the  web  of  the  air-pump  crosshead  in  this 
engine. 

TO  FIND  THE  PROPER  DEPTH  OF  THE  WEB  OF  THE  AIE-PUMP 
OROSSHEAD  AT  JOURNALS. 

KULE. — Mutiply  the  diameter  of  the  cylinder  in  inches  ~by  "061. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '061  =  2*44  inches,  which  is  the  proper 
depth  of  the  web  of  the  air-pump  crosshead  at  the  journals  in 
this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 


DIMENSIONS    OF   AIR-PUMP   CROSSHEAD.  283 

Then  64  inches  x  '061  =  3'90  inches,  which  is  the  proper 
depth  of  the  weh  of  the  air-pump  crosshead  at  the  journals  in 
this  engine. 

TO   FIND   THE   PROPER  THICKNESS   OF   THE   EYE   OF   THE   AIR- 
PUMP   OEOSSHEAD. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  by  '025. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '025  =  I'OO  inches,  which  is  the  proper 
thickness  of  the  eye  of  the  air-pump  crosshead  in  this  engine. 

Example  2. — Let  64  inches  he  the  diameter  of  the  cylinder. 

Then  64  inches  x  -025  =  1'600  inches,  which  is  the  proper 
thickness  of  the  eye  of  the  air-pump  crosshead  in  this  engine. 

TO   FIND   THE   PROPER   THICKNESS    OF   THE   WEB    OF   THE  AIR-PUMP 
CEOSSHEAD   AT   THE  MIDDLE. 

KULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  -043. 

Example  1. — Lot  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '043  =  1*72  inches,  which  is  the  proper 
thickness  of  the  web  of  the  air-pump  crosshead  at  the  middle 
in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '043  =  2-75  inches,  which  is  the  proper 
thickness  of  the  web  of  the  air-pump  crosshead  at  the  middle 
in  this  engine. 

TO   FIND  THE  PROPER  THICKNESS   OF  THE  WEB   OF   THE  AIR-PUMP 
CROSSHEAD  AT  THE  JOURNALS. 

EULE. — Multiply  the  diameter  of  the  cylinder  in  inches  "by  '037. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '087  =  1'48  inches,  which  is  the  proper 
thickness  of  the  web  of  the  air-pump  crosshead  at  the  journals 
in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '087  =  2-36  inches,  which  is  the  proper 
thickness  of  the  web  of  the  air-pump  crosshead  at  the  journals 
in  this  engine. 


284  PKOPOBTIONS   OF   STEAM-ENGINES. 

TO  FIND  THE  PEOPEE  DIAMETEE  OF  THE  JOUBNALS  OF  THE  AIE-PUMP 
CKOSSHEAD. 

RULE. — Multiply  tlio  diameter  of  the  cylinder  in  inches  ~by  '051. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '051  =  2'04  inches,  which  is  the  proper 
diameter  of  the  journals  of  the  air-pump  crosshead  in  this  en- 
gine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '051  =  3'26  inches,  which  is  the  proper 
diameter  of  the  journals  of  the  air-pump  crosshead  in  this  en- 
gine. 

TO  FIND  THE   PEOPEE   LENGTH  OF  THE   JOUENALS  OF  THE  AIB-PUMP 
OEOSSHEAD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  l>y  "058. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '058  =  2'32  inches,  which  is  the  proper 
length  of  the  end  journals  for  the  air-pump  crosshead  in  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '058  =  3'TL  inches,  which  is  the  proper 
length  of  the  end  journals  for  the  air-pump  crosshead  in  this 
engine. 

AIR-PUMP  SIDE  RODS. 

TO  FIND  THE  PEOPEE  DIAMETEE  OF  AIE-PUMP  SIDE  BOD  AT  THE  ENDS. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  5y  *039. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '039  =  1/56  inches,  which  is  the  proper 
diameter  of  air-pump  side  rod  at  ends  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '039  =  2*49  inches,  which  is  the  proper 
diameter  of  air-pump  side  rod  at  ends  in  this  engine. 

TO  FIND  THE  BEEADTH  OF  BUTT  FOE  AIE-PUMP  SIDE  BODS. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '046. 


DIMENSIONS   OF   AIR-PUMP   SIDE   RODS.  285 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '046  =  l'S4  inches,  which  is  the  proper 
breadth  of  butt  for  air-pump  side  rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '046  =  2'94  inches,  which  is  the  proper 
breadth  of  butt  of  air-pump  side  rod  in  this  engine. 

TO  FIND  THE  PBOPEB  THICKNESS  OF  BUTT  FOE  AIE-PUMP  SIDE  EOD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '037. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -037  =  1 '48  inches,  which  is  the  proper 
thickness  of  butt  for  air-pump  side  rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '037  =  2'36  inches,  which  is  the  proper 
thickness  of  butt  for  air-pump  side  rod  in  this  engine. 

TO  FIND   THE   MEAN  THICKNESS   OF  STRAP  AT  CUTTEB  OF  AIB-PUMP 
BIDE   EOD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  T>y  '019. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '019  =  -76  inches,  which  is  the  proper 
mean  thickness  of  the  strap  at  cutter  of  air-pump  side  rod  in 
this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  '019  =  1*21  inches,  which  is  the  proper 
mean  thickness  of  the  strap  at  cutter  of  air-pump  side  rod  in 
this  engine. 

TO  FIND  THE  PEOPEE  MEAN  THICKNESS  OF  THE  8TEAP  BELOW  OUT- 
TEE  OF  AIB-PUMP  SIDE  EOD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ly  '014. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  -014  =  -56  inches,  which  is  the  proper 
mean  thickness  of  the  strap  below  cutter  in  the  air-pump  side 
rod  of  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 


286  PROPORTIONS   OF   STEAM-ENGINES. 

Then  64  inches  x  '014  =  '89  inches,  which  is  the  proper 
mean  thickness  of  strap  below  cutter  in  the  air-pump  side  rod 
of  this  engine. 

TO   FIND   THE   PROPEB   DEPTH   OP  THE  GIBS  AST)   CUTTER   FOR  AIR- 
PUMP  SIDE  ROD. 

RULE. — Multiply  the  diameter  of  the  cylinder  in  inches  ~by  '048. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  x  '048  =  1'92  inches,  which  is  the  proper 
depth  of  gibs  and  cutter  for  air-pump  side  rod  in  this  engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  x  *048  =  3'OV  inches,  which  is  the  proper 
depth  of  gibs  and  cutter  for  the  air-pump  side  rod  in  this  engine. 

TO  FIND  THE  PROPER  THICKNESS  OF  THE  GIBS  AND  CUTTER  FOR  THE 
AIR-PUMP   SIDE   ROD. 

RULE. — Divide  the  diameter  of  the  cylinder  in  inches  ~by  100. 

Example  1. — Let  40  inches  be  the  diameter  of  the  cylinder. 

Then  40  inches  -5-  100  =  '40  inches,  which  is  the  proper 
thickness  of  the  gibs  and  cutter  for  the  air-pump  side  rod  in  this 
engine. 

Example  2. — Let  64  inches  be  the  diameter  of  the  cylinder. 

Then  64  inches  -5- 100  =  '64  inches,  which  is  the  proper 
thickness  of  the  gibs  and  cutter  of  the  air-pump  side  rod  in  this 
engine. 

It  will  be  satisfactory  to  compare  the  dimensions  of  the  parts 
of  engines  with  the  actual  dimensions  obtaining  in  some  engines 
of  good  proportions  which  have  for  some  time  been  in  success- 
ful operation ;  and  I  select  for  the  purpose  of  this  comparison 
the  side-lever  engines  constructed  by  Messrs.  Caird  &  Co.,  for 
the  West  India  Mail  steamers  '  Clyde,'  '  Tweed,'  '  Tay,'  and 
'  Tevoit.'  The  dimensions  of  the  main  parts  given  by  the  rules, 
and  the  actual  dimensions,  are  exhibited  in  the  following  table, 
touching  which  it  is  sufficient  to  remark  that  where  there  is  any 
appreciable  divergence  between  the  two,  the  dimensions  given 
by  the  rules  appear  to  be  the  preferable  ones : — 


RULES    TESTED    BY   PRACTICAL   EXAMPLES. 


287 


COMPABISON  OF  DIMENSIONS  GIVEN  BY  THE  FOREGOING  BtTLES 
WITH  THE  ACTUAL  DIMENSIONS  OF  THE  MAIN  PAETS  OF  THE 
SIDE  LEVEE  ENGINES  OF  THE  STEAMERS  '  CLYDE,'  '  TWEED,' 
'TA.Y,'  AND  'TEVIOT,'  OF  450  HOESES  POWER,  CONSTEUCTED 
BY  MESSES.  CAIED  &  CO. 


Diameter  of  Paddle-Shaft  Journal. 

Dimensions 
by  Rulea. 

Actual 
Dimensions. 

Diameter  of  paddle-shaft  journal  

15-15 

15-25 

Exterior  diameter  of  large  eye  of  crank  

27-84 

27-875 

10-49 

9-5 

Exterior  diameter  of  small  eye  of  crank  
tLength  of  small  eye  of  crank  

19-81T7 
13-875 

20-625 
13-25 

Thickness  of  web  of  crank  at  paddle  centre  
at  crank  pin  centre  
Breadth  of  crank  at  crank  pin  centre  

9'8 
8-14 
12-21 

10-5 
9-75 
15-0 

Diameter  of  piston  rod  

7-4 

7-75 

7-03 

7-6 

9-98 

9-25 

side  rod  at  ends  

4-77 

5-0 

u                      "                 at  middle  

6-6 

6-375 

"              ey  o  of  crosshead  (outside)  

13-5 

14-5 

Depth  of  eye  of  crosshead  (outside)  

21-138 

21-25 

Diameter  of  crosshead  journal  

6-349 

6-375 

Thickness  of  web  of  crosshead  at  centre  

5-8 

5-5 

tDepth  of  web  of  crosshead  at  centre  

19-85 

19-5 

4-514 

4-875 

Depth  of  web  of  crosshead  at  journal  

7-511 

9-75 

tDiameter  of  main  centre  journal  < 

0-0367  x 
P*xD= 

13-579 

I     11-5 

The  rules  give  generally  smaller  numbers  than  Messrs.  Caird's 
practice.  The  difference  is  greatest  in  'Breadth  of  crank  at 
crank-pin  centre,'  and  in  '  Exterior  diameter  of  eye  of  crosshead,' 
and  'Depth  of  web  of  crosshead  of  journals.' 

In  five  cases  above,  marked  thus  t,  the  rules  give  greater 
strength  than  the  example  selected  of  Messrs.  Caird's  engine, 
especially  in  '  Diameter  of  main  centre,'  where  Messrs.  Caird's 
proportions  are  quite  too  small. 

I  have  already  explained  that  from  any  one  drawing,  all  sizes 
of  engines  of  that  particular  form  may  be  constructed  by  merely 
altering  the  scale ;  and  all  the  dimensions  of  ships  and  engines, 
and,  in  fact,  every  quantity  whatever  which  increases  or  dimin- 
ishes in  a  given  ratio,  or  according  to  a  uniform  law,  may  be  ex- 
pressed graphically  by  a  curve,  which  will  have  its  correspond- 
ing equation,  though  sometimes  that  equation  will  be  too  com 
plicated  to  be  numerically  expressible.  Mr.  "Watt,  in  his  early 


288  PROPORTIONS   OP   STEAM-ENGINES. 

practice,  laid  down  most  of  the  dimensions  of  his  engines  to 
curves,  and,  indeed,  was  in  the  habit  of  using  that  mode  of  in- 
vestigation and  expression  in  all  his  researches.  A  table  of  the 
dimensions  of  the  parts  of  engines  may  easily  be  laid  down  in 
the  form  of  a  curve ;  and  the  benefit  of  that  practice  is,  that  if 
we  have  a  certain  number  of  points  in  the  curve,  we  can  easily 
find  all  the  intermediate  ones  by  merely  measuring  with  a  pair 
of  compasses  and  a  scale  of  equal  parts.  Thus,  for  example,  we 
may  lay  down  the  table  of  the  diameter  of  crank-shaft,  given  in 
page  294,  to  a  curve  as  follows : — First  draw  a  straight  horizon- 
tal line,  which  divide  into  equal  parts  by  any  convenient  scale, 
beginning,  as  in  the  table,  with  20,  and  ending  with  100.  If  now 
we  erect  vertical  lines  or  ordinates  at  every  division  of  the  hori- 
zontal line,  and  if,  with  any  given  length  of  stroke,  say  2  feet, 
we  know  the  diameter  of  shaft  proper  for  some  of  the  diameters 
of  cylinder — say  for  a  20-inch  cylinder,  4*08  inches ;  for  a  24-inch 
cylinder,  4'66  inches;  for  a  40-inch  cylinder,  6'55  inches;  and 
for  an  80-inch  cylinder,  10*29  inches — we  can  easily  determine 
the  diameters  of  shaft  proper  for  all  the  intermediate  diameters 
of  cylinders,  by  marking  off  with  the  same  scale,  or  any  other, 
the  vertical  heights  corresponding  to  all  the  diameters  we  know ; 
and  a  curve  traced  through  these  points  will  intersect  all  the 
other  ordinates,  and  give  the  diameters  proper  for  the  whole 
series.  By  thus  setting  down  the  known  quantities  in  order  to 
deduce  the  unknown,  we  shall  at  the  same  time  see  whether  the 
quantities  we  set  down  follow  a  regular  law  of  increase  or  not ; 
for  if  they  do  not,  instead  of  all  the  points  marked  off  falling 
into  a  regular  curve,  some  of  them  will  be  above  the  curve  and 
some  of  them  beneath  it,  thus  showing  that  the  quantities  given 
do  not  form  portions  of  a  homogeneous  system.  If  the  quantity 
increases  in  arithmetical  progression,  the  curve  will  become  a 
straight  angular  line.  Thus  in  the  case  of  the  diameter  of  the 
piston  rod,  as  the  increase  follows  the  same  law  as  the  increase 
of  the  diameter  of  the  cylinder,  the  law  of  increase  will  be  ex- 
pressed by  a  right-angled  triangle,  the  diameters  of  the  cylinder 
being  represented  by  the  divisions  on  the  base,  and  the  diameters 
of  piston  rod  by  the  corresponding  vertical  ordinates.  If  to  the 


KULES    TESTED    BY   PKACTICAL   EXAMPLES.  289 

curve  of  diameter  of  crank  shaft  for  each  diameter  of  cylinder 
with  any  given  length  of  stroke,  we  add  below  the  base  another 
curve  pointing  downwards,  representing  the  increase  of  the  di- 
ameter of  the  shaft  due  to  every  increase  of  the  length  of  the 
stroke,  the  diameter  of  the  cylinder  remaining  the  same,  the  total 
height  of  the  conjoint  ordinates  will  show  the  diameter  of  the 
shaft  for  each  successive  diameter  of  cylinder  and  length  of 
stroke.  One  of  the  curves  will  be  convex  and  the  other  con- 
cave, and  the  convexity  of  the  one  will  be  equal  to  the  concavity 
of  the  other,  so  that  the  ordinates  will  be  the  same  as  those  of  a 
triangle.  Hence,  if  we  double  the  diameter  of  the  cylinder,  and 
also  double  the  length  of  the  stroke,  we  shall  double  the  diam- 
eter of  the  shaft ;  if  we  treble  the  diameter  of  the  cylinder,  and 
also  treble  the  length  of  the  stroke,  we  shall  treble  the  diameter 
of  the  shaft,  and  so  on  in  all  other  proportions.  By  referring  to 
the  table  in  page  294,  we  shall  see  that  these  relations  are  there 
preserved.  Thus  a  20-inch  cylinder  and  a  2-feet  stroke  has  a 
shaft  of  4'08  inches  in  diameter ;  a  40-inch  cylinder  and  a  4-feet 
stroke,  a  shaft  of  8'16  inches  diameter;  a  60-inch  cylinder  and 
a  6-feet  stroke,  a  shaft  12'25  inches  diameter,  and  so  on.  If  this 
were  not  so,  an  engine  drawn  on  any  one  scale  would  not  be  ap- 
plicable to  any  other  of  a  different  size ;  whereas  we  know  that 
any  one  drawing  will  do  for  all  sizes  of  engines  by  merely  chang- 
ing the  scale. 

It  is  very  convenient  in  making  drawings  of  engines  to  adopt 
some  uniform  size  for  the  drawing-boards  and  drawings,  and  to 
adhere  to  them  on  all  occasions.  The  best  arranged  drawing- 
office  I  have  met  with  is  that  of  Boulton  and  Watt,  which  was 
originally  settled  in  its  present  form  by  Mr.  Watt  himself,  who 
brought  the  same  good  sense  and  habits  of  methodical  arrange- 
ment to  this  problem  that  he  did  to  every  other.  The  basis  of 
Boulton  and  Watt's  sizes  of  drawings  is  the  dimensions  of  a  sheet 
of  double  elephant  drawing  paper;  and  all  their  drawings  are 
either  of  that  size,  of  half  that  size,  or  of  a  quarter  that  size, 
leaving  a  proper  width  for  margin.  The  drawing-boards  are  all 
made  with  a  frame  fitting  around  them,  so  that  it  is  not  neces- 
sary to  glue  the  paper  round  the  edges;  but  the  damped  sheet 
13 


290 


PROPORTIONS   OF   STEAM-ENGINES. 


TABLE   OF   THE  DIMENSIONS   OF  THE  PBINCIPAL 
MARINE  ENGINES   OF 


NAMES  OF  PABTS. 

NOMINAL 

PH' 

rf 

o 

PM 

rf' 

o 

(H 

w 

o 
e* 

PL,' 
H 
S 

Diameter  of 
Cylinder  

in. 
20 
2 
12 
If 

n 

2i 
1* 
4 
5 
8* 
2 
li 
2* 

9 

2 

24 
12 
6 

6 
4 
5 
11 

4* 
2* 
8} 

4 
It 

29} 
83 
21 
66 

» 

i} 

2 
13 

14 
5 
1 

in. 
24 
21 
15 
If 
1} 
2} 

? 

6 
4} 
2£ 

P 

5* 
11 
2* 

80 
15 

7} 

7* 
4} 
6* 

in. 

27 

1? 

2 

3lf 
2 

? 

5 
2} 

I 

6} 
11 
2* 

80 
15 
71 

8 
6 
6} 
1J 

5J 
8* 
4i 

H 

5J- 
6} 

8T} 

42i 
25} 
76 
10 
2 

2i 
15* 

19 
6} 

fl 

in. 
29} 
8 
17} 

I 

8J 
? 

P 
1 

I 

2} 

83 
16} 
8 

9 

5} 

2 

6} 
4 
5 
If 

5} 
6f 

89} 
45 
26 
80 
11 
2} 

2} 
17 

21 

jf 

Piston  rod  

Air-pump  

Air-pump  rod.  

Steam-pipe  

Waste-water  pipe  

Beam  gudgeon  

Crank-pin  

Main  shaft  

Paddle-wheels,  in  feet  

Weight-shaft  bearings  

Stroke  of 
Piston  

Air-pump  bucket  

Feed-pump  plunger  

Cylinder  cross/lead 
Depth  of  boss  

Depth  of  middle  

Air-pump  crosshead 
Depth  of  boss  

5 

? 
1| 

4* 
5J 

84} 
39 
23 
72 
81 
U 

2 
14 

18 
* 

Thickness      

Columns 
Diameter  at  top  

Diameter  at  bottom  

Centre  to  centre  of 

fort  valve  passage 
Depth  

Width  

Seam 
Breadth  at  middle  

Breadth  at  ends  

Thickness  

EXAMPLES    OF    APPROVED    DIMENSIONS. 


291 


PABTS   OF   MESSES.   MATTDSLAY,    SONS,    AND  FIELD  8 
DIFFERENT   POWEES. 


POTTEB  OP  ENGLSE. 

ft 
§ 

Pi 

ft 
3 

W 
2 

Pi 

ft 

S 

Pi 

0 

H 

PL,' 

ft 

a 

0 

Pi 

w 

o 

TH 
1-1 

Pi 

S 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

82 

86} 

40 

43 

46 

48 

50 

52} 

55} 

57 

3} 

3* 

4 

4} 

4* 

41 

4J 

5 

5} 

5* 

21 

23 

24 

26 

28 

80 

81} 

84 

2} 

2* 

21 

25 

3 

3} 

8* 

81 

4 

4} 

2 

2} 

2* 

21 

3 

3} 

8} 

8J 

8* 

8* 

4 

4} 

4* 

5 

4 

6 

6* 

7 

7* 

2} 

2* 

2* 

21 

8 

8} 

8} 

3* 

8* 

4 

6* 

7 

71 

8* 

9} 

10 

10* 

11 

11* 

12 

8 

9 

9* 

10 

11* 

12} 

18 

13* 

14 

5* 

6 

6* 

7 

7* 

8 

8* 

9 

9} 

9} 

3} 

8* 

4 

4} 

4* 

ft 

5 

5} 

6} 

5* 

2 

2} 

21 

2* 

3 

4 

8} 

.     4 

4* 

5 

5* 

6* 

6* 

7 

7f 

7} 

8 

7 

8* 

9} 

10 

10} 

10* 

11* 

12 

12} 

18 

18 

15 

17 

17 

19 

19 

21 

21 

23 

2f 

2} 

21 

21 

3 

8} 

3} 

8* 

81 

8} 

86 

36 

42 

48 

52 

56 

60 

63 

66 

72 

18 

13 

21 

24 

26 

28 

80 

31* 

88 

86 

9 

9 

10* 

12 

13 

14 

15 

16 

16* 

18 

9* 

10* 

12 

13 

14 

14* 

15 

16 

17 

17} 

6 

61 

7* 

8 

81 

9 

9* 

10 

11 

12 

7} 

8* 

9* 

10} 

11* 

Hi 

12* 

18 

13* 

14 

2} 

2* 

21 

8 

3} 

8* 

3* 

3} 

4f 

61 

8 

9 

10 

10* 

10} 

11 

H* 

12 

12* 

% 

8 

7* 

5} 

61 

8 

6 
8} 

6* 

8* 

9* 

S 

7* 
9* 

1* 

H 

2 

2s 

2} 

2} 

2* 

21 

21 

6 

7 

8 

8} 

91 

9} 

9* 

10 

10} 

10* 

61 

ft 

9 

10* 

lot 

11 

11* 

"I 

12 

42} 

47* 

53 

551 

60} 

63 

67 

68} 

70 

72 

48 

54 

60 

63 

69 

69 

72 

78 

80 

88 

27 

80 

84 

84 

40 

40 

42 

44 

45 

46 

84 

83 

96 

100 

108 

108 

112 

126 

128 

180 

11* 

18 

15 

18* 

18* 

19 

19 

20 

20 

21 

21 

3 

8 

4 

4 

4} 

4* 

41 

4} 

3} 

81 

4 

4* 

5 

5} 

5* 

6 

6* 

7 

18 

20 

24 

26 

28 

28 

29 

81 

81 

32 

28 

25 

28 

29 

83 

34 

85 

86 

88 

89 

8 

81 

10 

10* 

12 

12} 

12} 

12 

15 

15* 

1* 

H 

H 

2 

2} 

2f 

2* 

2* 

2* 

21 

292 


PROPORTIONS   OF   STEAM-ENGINES. 


TABLE   OF   THE   DIMENSIONS   OF   THE   PRINCIPAL 
ENGINES    OF    DIF- 


— 

.5 

.4  « 

3 

•i 

•s 

| 

i| 

id 

•9 

£•» 

w^s 

1 

a 

3  . 

'S, 

*  ^ 

a  g 

• 

I 

*& 

$£ 

S-o 

13  1 

It 
II 

! 

(0 

^-—  ' 

°M 

I* 

i 

°H 
1-i 

a 

fl 
•3% 

F 

"3  § 

I  0, 

•5  s 

H 

if 

*& 
I* 

1 

& 

i 

p* 

s 

5 

Q 

5 

fig. 

a* 

5 

ft.  in. 

ft. 

in. 

in. 

in. 

in. 

in. 

in. 

10 

20 

12 

2  0 

9 

4 

7 

2 

li 

1 

4 

15 

24 

15 

2  6 

11 

4} 

8* 

2* 

1* 

li 

5 

20 

27 

17 

2  6 

11 

6f 

10 

3 

If 

1* 

5} 

30 

31} 

IS* 

3  0 

13 

6f 

10} 

3} 

If 

If 

6* 

40 

3G* 

20 

3  0 

13 

7* 

11* 

3* 

2} 

11 

7 

50 

39} 

22 

3  6 

15 

8} 

12} 

3} 

2* 

2 

7* 

60 

43 

24 

4  0 

17 

9 

13 

4 

2} 

21 

8* 

70 

46 

25* 

4  3 

17 

8| 

13} 

41 

3 

2* 

9} 

80 

48 

27 

4  6 

19 

10 

14* 

4* 

81 

2} 

10 

90 

50 

23 

4  9 

19 

10* 

15} 

4} 

8* 

8 

10* 

100 

52* 

5  0 

21 

11 

16 

5 

8f 

Si 

11 

110 

55 

5  0 

21 

11* 

16} 

5} 

4 

81 

11* 

120 

57* 

5  6 

23 

12* 

17* 

5* 

41 

8* 

12 

is  laid  upon  the  board,  which  it  somewhat  overlaps,  and  the 
frame  then  comes  down  and  turns  over  the  edges  of  the  paper 
upon  the  sides  of  the  board,  and  the  frame  being  then  fixed  so 
that  its  face  is  flush  with  the  paper,  the  paper  by  being  thus 
bound  all  round  the  edges  is  properly  stretched  when  dry.  In 
Mr.  Watt's  time  the  drawings  were  made  with  copying  ink,  and 
an  impression  was  taken  from  them  by  passing  them  through  a 
roller  press,  so  as  to  retain  the  original  in  the  office,  while  a 
duplicate  of  it  was  sent  out  with  the  work ;  and  the  copying 
press  was  invented  by  Mr.  Watt  for  this  purpose.  The  whole 
of  the  drawings  pertaining  to  each  particular  engine  are  placed 
in  a  small  paper  portfolio  by  themselves ;  and  these  small  port- 
folios are  numbered  and  arranged  in  drawers,  with  a  catalogue 
to  tell  the  particular  engine  delineated  in  the  drawings  of  each 
portfolio.  In  this  way  I  have  found  that  the  drawings  illustra- 
tive of  any  engine,  though  it  may  have  been  made  in  the  last 


EXAMPLES    OF   APPROVED   DIMENSIONS. 


293 


PARTS   OF  MESSES.   SEAWARD   AND   CQ.'s   MARIXE 
FEREXX   POWERS. 


I 

i 

, 

•O 

to 

• 

M 

j 

II 

1 

M 

|| 

. 
If 

•c 

•s  J 

if 

1 

1 

s  J 

l-s 

|4 

o 

5, 

to-2 

S  a 

11 

ii 

1^ 

"S 

s  i 

i 

bo 

JS3, 

*&= 

II 

J 

3 

J 

a 

i 

i 

*f 

|. 

S 

J 

in. 

in. 

ft.  in. 

in. 

in. 

in. 

in. 

in. 

in. 

In. 

26 

5 

1* 

6  0 

2 

8 

1§ 

2* 

2 

2* 

2f 

28 

6 

H 

7  0 

21 

4 

1* 

2* 

21 

3 

31 

30 

7 

2 

8  0 

2* 

5 

It 

3 

2* 

8* 

3J 

85 

8 

8i 

8  8 

2f 

5* 

2 

3* 

2* 

4 

41 

38 

9 

2* 

10  0 

3 

6 

21 

3* 

2} 

4* 

41 

40 

9* 

2t 

10  6 

81 

e* 

2* 

3i 

2f 

5 

M 

44 

10 

8 

11  6 

8* 

7 

3f 

4 

2J 

5* 

6 

60 

10 

31 

12  6 

8} 

7* 

3 

41 

8 

6 

6* 

66 

11 

3* 

13  0 

4 

8 

31 

4* 

3* 

6* 

7 

68 

12* 

3i 

13  6 

41 

8* 

3* 

4t 

81 

7 

7* 

62 

13 

4 

16  0 

4* 

9 

3| 

5 

3* 

7| 

7* 

66 

18* 

41 

16  0 

4t 

9* 

8t 

51 

3t 

7t 

8i 

70 

14 

4* 

17  6 

5 

9J 

35 

5* 

3t 

8 

8* 

century,  could  be  produced  to  me  in  a  few  minutes ;  and  the 
system  is  altogether  more  perfect  and  more  convenient  than  any 
other  with  which  I  am  acquainted.  The  portfolios  are  not  large, 
which  would  make  them  inconvenient,  but  are  of  such  size  that 
a  double  elephant  sheet  has  to  be  folded  to  go  into  one  of  them ; 
but  most  of  the  drawings  are  on  small  sheets  of  paper,  which  is 
a  much  more  convenient  practice  than  that  of  drawing  the  de- 
tails upon  large  sheets. 

It  will  be  interesting  to  compare  with  the  results  given  in 
the  foregoing  rules  the  actual  sizes  of  some  side  lever  engines  of 
approved  construction.  Accordingly  I  have  recapitulated,  in  the 
tables  introduced  above,  the  principal  dimensions  of  the  marine 
engines  of  Messrs.  Maudslay  and  Messrs.  Seaward.  These  tables 
ore  so  clear,  that  they  do  not  require  further  explanation,  and 
the  same  remark  is  applicable  to  the  tables  which  follow. 


294 


PROPORTIONS   OF   STEAM-EXGINES. 


DIAMETER   OF  WEOTJGHT-IEON   CKAXK-SHAFT     JOUKNAL. 


~  Ji 

3-r3 

jo 

LENGTH   OF   STROKE    IN   FEET. 

2 

2.L        3 

3^ 

4 

^ 

5 

^ 

6 

7 

8 

9 

20 

4-03   !   4-39 

4-67 

4-91 

5-14 

5-34 

5-53 

5-72 

5-83 

6-19 

6-43 

6-73 

21 

4-23 

4-54 

4-82 

5-05 

5-30 

5-50 

5-71 

5-89 

6-07 

6-39 

6-63 

6-95 

22 

4-37 

4-63 

4-9G 

5-20 

5-46 

5-66 

5-83 

6-07 

6-25 

6-53 

6-83 

7-1G 

23 

4-52 

4-81 

5-11 

5-35 

5-G2 

5-83 

6-05 

6-25 

6-43 

6-77 

7-08 

7-37 

24 

4-66 

4-95 

5-25 

5-51 

5-73 

5-99 

6-22 

6'43 

6-62 

6-96 

7-23 

7-58 

25 

4-31 

5-09 

5-40 

5-60 

5-94 

6-16 

6-40 

6-60 

6-80 

7-15 

7-49 

7-78 

26 

4-95 

5-22 

5-54 

5-81 

6-10 

6-33 

6-57 

6-78 

6-98 

7-35 

7-69 

7-99 

27 

510 

5-36 

5-69 

5-9G 

6-26 

6-49 

6-74 

6-96 

7-16 

7-54 

7-89 

8-20 

28 

5-24 

5-49 

5-83 

6-11 

6-42 

6-66 

6-91 

7-14 

7-35 

7-73 

8-09 

8-41 

29 

5-33 

5-53 

5-98 

6-26 

6-53 

6-32 

7-08 

7-31 

7-53 

7-92 

8-29 

8-62 

80 

5-42 

5-67 

6-12 

6-42 

6-74 

6-99 

7-26 

7-49 

7-72 

8-12 

8-49 

8-83 

31 

5-47 

5-88 

6-25 

6-56 

6-34 

7-14 

7-41 

7-65 

7-88 

8-29 

8-67 

9-02 

32 

5-59 

6-00 

6-88 

6-69 

6-93 

7-29 

7-56 

7-81 

8-04 

8-46 

8-85 

9-21 

33 

5-71 

6-12 

6-51 

6-83 

7-12 

7-43 

7-71 

7-97 

8-20 

8-63 

9-03 

9-39 

34 

5-83 

6-24 

6-64 

6-96 

7-26 

7-53 

7-37 

8-13 

S-37 

8-30 

9-21 

9-58 

35 

5-95 

6-37 

6-77 

7-10 

7-41 

7-73 

8-02 

8-28 

8-53 

3-97 

9-39 

9-76 

36 

6-07 

6-49 

6-90 

7-23 

7-55 

7-83 

8-17 

8-44 

8-69 

9-14 

9-57 

9-95 

87 

6-19 

6-61 

7-03 

7-37 

7-69 

8-03 

8-32 

8-60 

8-85 

9-31 

9-75 

10-14 

38 

6-31 

6-73 

7-16 

7-50 

7-33 

8-17 

8-47 

8'76 

9-02 

9-48 

9-93 

10-33 

39 

6-43 

6-85 

7-29 

7-64 

7-97 

8-32 

8-63 

8-91 

9-18 

9-65 

10-11 

10-51 

40 

6-55 

6-93 

7-42 

7-78 

8-16 

8-47 

8-79 

9-07 

9-35 

9-38 

10-29 

10-70 

41 

6-57 

7-19 

7-54 

7-90 

8-29 

8-60 

8-93 

9-21 

9-50 

9-99 

10-45 

10-87 

42 

6-67 

7-20 

7-66 

8-03 

8-42 

8-74 

9-07 

9-36 

9-65 

10-15 

10-62 

11-04 

43 

6-77 

7-31 

7-78 

8-15 

8-55 

8-87 

9-21 

9-50 

9-80 

10-31 

10-78 

11-21 

44 

6'87 

7-42 

7-90 

8-28 

8-63 

9-01 

9-35 

9'65 

9-95 

10-47 

10-95 

11-38 

45 

6-97 

7-54 

8-02 

8-40 

8-81 

9-14 

9-50 

9-79 

10-10 

10-63 

11-11 

11-55 

46 

7-06 

7-65 

8-15 

8-53 

8-94 

9-28 

9-64 

9-94 

10-25 

10-78 

11-28 

11-72 

48 

7-26 

7-88 

8-40 

8-78 

9-20 

9-55 

9-92 

10-23 

10-54 

11-09 

11-61 

12-06 

50 

7-48 

8-10 

8-61 

9-02 

9-46 

9-82 

10-20 

10-51 

10-84 

11-42 

11-94 

12-41 

52 

7-70 

8-31 

8-83 

9-26 

9-71 

10-08 

10-47 

10-79 

11-12 

11-71 

12-25 

12-78 

54 

7-90 

8-52 

9-05 

9-50 

9-96 

10-34 

10-74 

11-06 

11-40 

12-00 

12-56 

13-05 

56 

8-09 

8-73 

9-27 

9-73 

10-21 

10-60 

11-01 

11-33 

11-69 

12-29 

12-87 

13-37 

58 

8-29 

8-94 

9-50 

9-97 

10-45 

10-86 

11-28 

11-61 

11-97 

12-58 

18-18 

18-69 

60 

8-49 

9-15 

9-72 

10-20 

10-70 

11-12 

11-55 

11-89 

12-25 

12-89 

18-48 

14-02 

62 

8-67 

9-35 

9-93 

10-40 

1089 

11-34 

11-79 

12-14 

12-51 

18-17 

13-77 

14-82 

64 

8-86 

9-55 

10-14 

10-60 

11-08 

11-56 

12-03 

12-39 

12-78 

13-45 

1406 

14-62 

66 

9-04 

9-74 

10-35 

10-79 

11-27 

11-78 

12-28 

12-64 

13-04 

18-73 

14-85 

14-98 

68 

9-22 

9-94 

1056 

10-99 

11-45 

11-99 

12-52 

12-89 

18-30 

14-01 

14-64 

15-28 

70 

9-41 

10-14 

10-77 

11-19 

11-64 

12-20 

12-77 

18-16 

13-57 

14-29 

14-94 

15-54 

72 

9-59 

LO-33 

10-97 

11-41 

11-90 

12-44 

13-01 

13-40 

13-82 

14-55 

15-22 

15-83 

74 

9-76 

10-52 

11-17 

11-62 

12-16 

12-67 

13-25 

18-64 

14-07 

14-82 

15-50 

1612 

76 

9-93 

10-71 

11-36 

11-83 

12-43 

12-91 

13-49 

13-89 

14-32 

15-09 

15-78 

16-41 

78 

10-11 

10-89 

11-56 

12-05 

12-69 

13-15 

13-78 

14-13 

14-57 

15-85 

16-06 

16-70 

80 

10-29 

11-08 

11-76 

12-27 

12-96 

13-38 

13-96    14-38 

14-84 

15-62 

16-33 

16-98 

82 

10-46 

11-26 

11-96 

12-49 

13-17 

13-61 

14-19    14-62 

15-08 

15-88 

16-59    17-26 

84 

10-63 

11-44 

12-15 

12-71 

18-38 

13-84 

14-42 

14-85 

15-32 

16-18 

16-86 

17-54 

86 

10-80 

11-61 

12-35 

12-92 

13-59 

14-07 

14-65 

15-09 

15.56 

16-38 

17-13 

17-82 

83 

10-97 

11-79 

12-54 

13-14 

13-80 

14-80 

14-88 

15-82  115-80 

16-63 

17-40 

18-10 

90 

11-13 

11-99 

12-74  ;13-34 

14-00 

14-54 

15-10    15-56    16-05 

16-89 

17-66 

18-37 

92 

11-29 

12-17 

12-92  13-47  114-21    14-75 

15-32    15-79 

16-28 

17-14 

17-92    18-64 

94 
96 

11-45 
11-61 

12-34 
12-51 

13-10    13-60 
13-29    13-73 

14-42 
14-63 

14-96 

15-18 

15-54    16-02 
15-76  !  16-25 

16-51 
16-74 

17-38 
17-63 

18-18 
18-44 

18-91 
19-18 

98 

11-77 

12-68 

13-47    13-86    14-84 

15-39 

15-98 

16-48 

16-97 

17-87 

18-70 

19-45 

100 

11-93 

12-85 

13-66 

14-01 

15-04 

15-61 

16-20 

16-71 

17-22    18-12 

18-95 

19-71 

TABLES   OF  NOMINAL   POWERS   OF   ENGINE.  295 

LENGTH   OF   WEOUGHT-IEOX   CBANK-SHAFT    JOUTCNAL. 


SSli 

i'2-a 
S=M 
B° 

LENGTH 

OP   STROKE    IN   FEET. 

2 

2i 

3 

3* 

4 

4* 

5 

5* 

6 

7 

8 

9 

20 

5-10 

5-49 

5-84  j     6-15     6-43 

6-69     6-93 

7-16 

7-;:'; 

7-75 

8-11 

S-43 

21 

5-37 

5-69 

6-03     6-31     6-62 

6-87     7-14 

7-36 

7-59 

7-99      8-86 

8-69 

22 

5-63 

5-85 

6-21      6-50;     6-82 

7-03     7-35 

7-59 

7-82 

8-23 

8-61 

8-96 

23 

5-68 

6-02 

6-39      6-69     7-02 

7-29     7-57 

7-S1 

8-05 

8-47 

8-86 

9-23 

24 

5-84 

6-19 

6-57     6-88     7-22 

7-50 

7-78 

8-04 

8-28 

8-71 

9-11 

9-50 

26 

5-99 

6-36 

6-75     7-07;    7-42 

7-70 

8-00 

8-26 

851 

8-95 

9-36 

9-76 

26 

6-15 

6-53 

6-931     7-26 

7-62 

7-91 

821 

8-48 

8-74 

9-19 

9-61 

10-02 

27 

6-30 

6-TO 

7-11 

7-45 

7-82 

8-11 

8-43 

8-70 

8-97 

9-43 

9-86 

10-28 

23 

6-46 

6-S7 

7-29 

7-64 

8-02 

8-32 

8-64 

8-92 

9-20 

9-67 

10-11 

10-54 

29 

6-61 

7-04 

7-47 

7-83 

8-22 

8-53 

8-S6 

9-14 

9-43 

9-91 

10-36 

10-SO 

80 

6-77 

7-21 

7-65 

8-02 

8-42 

8-74 

9-07 

9-36 

965 

10-15 

10-61 

11-04 

81 

6-92 

7-37 

7-82 

8-19 

8-60 

8-93 

9-27 

9-56 

9-86 

10-37 

10-84 

11-28 

82 

7-06 

7-52 

7-98 

8.36 

8-73 

9-11 

9-47 

9-76 

10-06 

10-58 

11-07 

11-52 

83 

7-20 

7-67 

8-14 

S-53 

8-96 

9-30 

9-56 

9-96 

10-27 

10-79 

11-30 

11-76 

84 

7-84 

7-82 

6-30 

8-70 

9-14 

9  -48 

9-75 

10-15 

10-47 

11-00 

11-53 

11-99 

35 

7-48 

7-97 

8-46 

8-87 

9-31 

9-67 

9-94 

10-35 

10-68 

11-22 

11-76 

12-22 

86 

7-62 

8-12 

8-63 

9-04 

9-49 

9-85 

1013 

10-55 

10-SS 

11-43 

11-98 

12-45 

87 

7-76 

8-27 

8-79 

9-21 

9-67 

10-04 

1082 

10-74 

11-09 

11-64 

12-20 

12-68 

83 

7-90 

842 

895 

9-38 

9-85 

10-22 

10-51 

10-94 

11-29 

11-86 

12-42 

12-91 

89 

8-05 

8-57 

9-11 

9-55 

10-03 

10-41    10-SO    11-14 

11-49 

12-07 

12-64 

13-14 

40 

8-19 

8-72 

9-27 

9-72 

10-20 

10-53 

10-99,  11-34 

11-69 

12-29 

12-86 

13-37 

41 

8-32 

8-86 

9-42 

9-88 

10-37 

10-75 

11-17    11-53 

11-88 

12-49 

13-07 

13-59 

42 

8-44 

9-00 

9-57 

10-03 

10-54 

10-92 

11-34    11-71 

1206 

12-69 

13-28 

13-81 

48 

8-56 

9-18 

9-72 

10-18 

10-70 

11-09 

11-52 

11-89 

1225 

12'S8 

13-49 

14-03 

44 

8-68 

9-27 

9-87 

10-37 

10-86 

11-26 

11-69 

12-07 

12-43 

13-03 

13-70 

14-25 

45 

8-80 

9-40 

10-02 

10-49 

11-02 

H-43 

11-87 

12-25 

12-62 

18-28 

13-91 

14-46 

46 

8-92 

9-54 

10-17 

10-64, 

11-18 

11-59 

12-04 

12-43 

12-80 

13-47 

14-12 

14-67 

43 

9-16 

9-81 

10-47 

10-94 

11-50 

11-93 

12-39 

12-79 

13-17 

13-87 

14-52 

15-09 

50 

9-40 

10-08 

10-76 

11-26 

11-82 

12-27 

12-75 

13-15 

13-55 

14-27 

1492 

15-51 

5-2 

9-65 

10-34 

1104 

11-56 

12-13 

12-59 

18-09 

13-50 

13-90 

14-64 

15-32 

15-92 

54 

9-89 

10-60 

11-82 

11-86 

12-44 

12-91 

13-43 

13-85 

14-25 

15-02 

15-71 

16-32 

56 

10-18 

10-86 

11-60 

12-16 

12-75 

13-23 

13-77 

14-19 

14-60 

15-39 

16-09 

16-72 

53 

10-87 

11-13 

11-88 

12-46 

13-06 

13-55 

14-11 

14-53 

14-95 

15-76 

16-47 

17-12 

60 

10-61 

11-37 

12-15 

12-75 

13-87 

18-87 

14-44 

14-86 

15-81 

16-11 

16-85 

17-52 

62 

10-84 

11-62 

12-41 

13-00 

13-61 

14-15 

14-76 

15-18 

15-64 

16-47 

17-23 

17-90 

64 

11-07 

11-86 

12-67 

18-25 

13-85, 

14-42    15-06 

15-50 

15-97 

16-83 

17-59 

1828 

66 

11-80 

12-11 

12-98 

18-50 

14-09 

14-70   15-86 

15-82 

16-80 

17-18 

1795 

18-66 

63 

11-58 

12-35   18-19 

18-75 

14-83 

14-97    15-66 

16-14 

16-63 

17-52 

18-31 

19-04 

70 

11-76 

12-60i  18-46 

14-01 

14-55 

15-25   15-96 

16-46 

16-96 

17-86 

18-67 

19-42 

72 

11-93 

12-84    13-71 

14-29 

14-88 

15-56i  16-26 

16-77 

17-28 

18-20 

19-03 

19-78 

74 

12-20 

18-07 

13-96 

14-58 

1521 

15-871  16-56 

17-08 

17-60 

18-54 

1939 

20-14 

76 

12-42 

18-80    14-20 

14-87 

15-54 

16-18  16-86 

17-89 

17-92 

18-87 

19-73 

20-50 

78 

12-64 

18-581  14-44    15-15 

15-87 

1649    17-16 

17-70 

18-24 

19-19 

20-07 

20-86 

80 

12-86 

13-76   14-70!  15-44 

16-20 

16-81  !  17-45 

18-00 

18-55 

19-52 

20-41 

21-22 

82 

18-07 

13-99    14-95 

15-70 

16-46 

17-08'  17-73 

18-29    18-86 

19-84 

20-75 

21-58 

84 

18-28 

14-22   15-19 

16-95 

16-72 

17-85   18-03 

18-58 

1916 

20-16 

21-09 

21-94 

86 

18-49 

14-45    15-48 

16-20 

16-98 

17-62    1881 

13-87 

19-46 

20-48 

21-43 

22-28 

88 

18-70 

14-68,  15-67    16-45 

17-24 

17-89    18-59 

19.16 

19-75 

20-80 

21-75 

22-62 

90 

38-91 

14-91    15-92    16-76 

17-50 

18-15   18-87 

19-47 

20-06 

21-11 

22-07 

22-96 

92 

14-11'  16-14   16-15   16-95 

17-76 

18-42    19-14 

19-75 

20-36 

2142 

22-40 

28-80 

94 

14-81    15-86    16-88 

17-19 

18-02 

18-69)  19-40 

20-01 

20-65 

21-73 

22-78 

23-64 

96 

14-51    15-59    16-61 

1743 

18-28 

18-96   19-67 

20-38 

20-94 

22-03 

2805 

28-98 

98 

14-71   15-82    16-84 

17-67 

18-54 

19-28    19'95 

20-57 

21-28 

2288 

28-37 

24-32 

100 

14-91    16-09    17-07 

17-92 

18-80 

19-52    20-25 

20-83 

21-52 

22-65 

28-69 

24-64 

i 

296 


PROPORTIONS   OF   STEAM-ENGINES. 


BREADTH    OF   WEB    OF    CRANK,  SUPPOSING   IT    TO    BE   CONTINUED 
TO    PADDLE-SHAFT    CENTRE. 


"S.S 
iJ-Sj 

LENGTH   OF   STROKE    IN   FEET. 

~6" 

2 

2i 

3 

3^ 

4 

4^ 

5 

B£ 

6 

7 

8 

9 

20 

5-28 

5-45 

5-62 

6-04 

6-46 

6-72 

6-98 

7-19 

7-40 

7-7S 

8-12 

8-44 

21 

5-51 

5-68 

5-S5 

6-27 

6-68 

6-94 

7-21 

7-42 

7-63 

8-04 

8-39 

8-71 

22 

5-74 

5-91 

6-03 

6-49 

6-90 

7-16 

7-44 

7-65 

7-86 

8-30 

8-65 

8-98 

23 

5-96 

6-14 

6-31 

6-72 

7-12 

7-38 

7-67 

7-88 

8-09 

8-55 

8-90 

9-25 

24 

6-18 

6-36 

6-54 

6-94 

7-33 

7-60 

7-89 

8-11 

8-32 

8-SO 

9-16 

9-52 

25 

6'40 

6-58 

6-77 

7-16 

7-54 

7-S2 

8-11 

8-34 

8-55 

9-04 

9-41 

9-79 

20 

6'62 

6-80 

7-00 

7-38 

7-75 

8-04 

8-33 

8-57 

8-78 

9-28 

9-67 

10-06 

27 

6-84 

7-03 

7-23 

7-60 

7-96 

8-26 

8-55 

8-80 

9-01 

9-51 

9-92 

10-33 

28 

7'06 

7-25 

7-46 

7-82 

8-17 

8-48 

8-77 

9-03 

9-24 

9-74 

10-18 

10-59 

29 

7-28 

7-48 

7-09 

8-04 

8-38 

8-70 

8-99 

9-26 

9-47 

9-97 

10-43 

10-85 

30 

7'50 

7-71 

7-92 

8-26 

8-00 

8-92 

9-24 

9-49 

9-74 

10-22 

10-68 

11-10 

31 

7-65 

7-88 

8-11 

8-46 

8-80 

9-13 

9-44 

9-72 

9-96 

10-44 

10-92 

11-34 

82 

7-80 

8-U5 

8-30 

8-66 

9-00 

9-34 

9-64 

9-94 

10-28 

10-67 

11-16 

11-58 

88 

7'95 

8-22 

8-49 

8-85 

9-20 

9-54 

9-S4 

10-15 

10-50 

10-89 

11-39 

11-82 

84 

8-10 

8-39 

8-68 

9-04 

9-40 

9-74 

1004 

10-35 

10-71 

11-12 

11-62 

12-06 

35 

8'25 

8-56 

8-87 

9-23 

9-59 

9-94 

10-24 

10-55 

10-92 

11-34 

11-85 

12-29 

36 

8-40 

8-78 

9-06 

9-42 

9-79 

10-14 

10-44 

10-75 

11-13 

11-56 

12-OS 

12-53 

3T 

8-55 

8-90 

9-25 

9-61 

9-99 

10-34 

10-64 

10-95 

11-34 

11-78 

12-31 

12-76 

83 

8-70 

9-07 

9-44 

9-80 

10-18 

10-54 

10-84 

11-15 

11-55 

12-00 

12-54 

12-99 

39 

8-85 

9-24 

9-62 

9-89 

10-37 

10-73 

11-04 

11-35 

11-76 

12-23 

12-77 

13-23 

40 

9-00 

9-40 

9-80 

10-18 

10-56 

10-90 

11-24 

11-56 

11-88 

12-46 

12-98 

13-43 

41 

9-19 

9-59 

9-99 

10-37 

10-75 

11-08 

11-41 

11-73 

12-05 

12-67 

13-22 

13-71 

42 

9-38 

9-78 

10-17 

10-56 

10-94 

11-25 

11-57 

11-89 

12-22 

12-87 

13-46 

13-94 

43 

9-57 

9-97 

10-36 

10-75 

11-18 

11-42 

11-72 

12-05 

12-39 

13-08 

18-70 

14-16 

44 

9-76 

10-15 

10-55 

10-94 

11-32 

11-59 

11-87 

12-21 

12-56 

13-28 

18-98 

14-38 

45 

9-94 

10-33 

10-74 

11-13 

11-51 

11-76 

12-02 

12-37 

12-78 

13-49 

14-17 

14-60 

46 

10-12 

10-51 

10-93 

11-82 

11-70 

11-98 

12-17 

12-53 

12-90 

13-69 

14-40 

14-82 

48 

10-48 

10-S7 

11-30 

11-70 

12-08 

12-27 

12-47 

12-85 

13-24 

14-10 

14-87 

15-26 

50 

10-S6 

11-27 

11-68 

12-07 

12-46 

12-61 

12-76 

13-17 

13-58 

14-54 

15-34 

15-70 

52 

11-20 

11-63 

12-06 

12-44 

12-84 

13-03 

13-22 

13-63 

14-04 

14-94 

15-71 

16-14 

54 

11-54 

11-98 

12-42 

12-81 

13-22 

13-45 

13-68 

14-09 

14-49 

15-34 

16-08 

16-56 

56 

11-88 

12-32 

12-78 

13-18 

18-60 

13-87 

14-15 

14-55 

14-95 

15-74 

16-45 

16-98 

58 

12-20 

12-66 

18-15 

18-56 

13-98 

14-29 

14-62 

15-01 

15-41 

16-14 

16-82 

17-41 

60 

12-52 

18-01 

13-51 

13-94 

14-87 

14-73 

15-10 

15-47 

15-84 

16-54 

17-19 

17-82 

62 

12-96 

13-41 

18-89 

14-82 

14-73 

15-10 

15-48 

15-85 

1  0-2-2 

16-92 

17-59 

18-23 

64 

13-38 

18-81 

14-26 

14-69 

15-09 

15-57 

15-86 

16-23 

16-60 

17-31 

17-99 

18-61 

66 

13-80 

14-21 

14-63 

15-05 

15-45 

15-94 

162-3 

16-61 

16-98 

17-70 

18-38 

19-05 

68 

14-22 

14-61 

15-00 

15-41 

15-81 

16-31 

16-60 

16-98 

17-36 

18-09 

18-78 

19-45 

70 

14-64 

15-01 

15-88 

15-77 

16-17 

16-57 

16-97 

17-35 

17-74 

18-48 

19-18 

19-85 

72 

15-02 

15-89 

15-75 

16-08 

16-56 

16-03 

17-30 

17-71 

18-12 

18-86 

19-58 

20-26 

74 

15-40 

15-77 

16-12 

16-46 

16-95 

17-29 

17-63 

18-06 

18-49 

19-24 

19-97 

20-67 

76 

15-78 

16-15 

16-49 

16-84 

17-34 

17-65 

17-97 

18-41 

18-86    19-62 

20-35  121-07 

78 

16-16 

16-58 

16-87 

17-22 

17-78 

18-01 

18-30 

18-76  J19-23    20-00 

20-73 

21-48 

80 

16-54 

16-89 

17-24 

17-68 

18-12 

18-37 

18-61 

19-11    19-61 

20-38  12111 

21-88 

82 

16-94 

17-20 

17-60 

18-04 

18-47 

18-75 

19-03 

19-51    19-99 

20-76 

21-51 

22-27 

84 

17-32 

17-65 

17-98 

18-40 

18-82 

19-18 

19-45 

19-90    20-36 

21-14 

21-89 

22-65 

86 

17-71 

18-03 

18-36 

18-76 

19-17 

19-51 

19-85 

20-29  '20.78    21-52 

22-28 

23-03 

88 

18-09 

18-41 

18-73 

19-12 

19-52 

19-89 

20-26 

20-68  ,21-10    21-89 

22-67 

•23-41 

90 

18-47 

18-79 

19-11 

19-49 

19-87 

20-27 

20-67 

21-07    21-47    22-26 

•23-03 

23-79 

92 

18-87 

19-18 

19-49 

19-87 

20-25 

20-64 

21-04 

21-44  121-84    22-64 

23-41 

24-17 

94 

19-26 

19-56 

19-87 

20-25 

20-62 

21-01 

21-41 

21  -SI    22-21  123-01 

23-79 

24-56 

96 

19-64 

19-94 

20-25 

20-63 

20-99 

21-38 

21-78 

22-18    22-48    23'88 

24-16 

•24-93 

98 

20-02 

20-82 

20-62 

21-00 

21-36 

21-75 

22-15 

22-55 

22-85    23-75 

24-53 

25-81 

100 

20-40 

20-70 

21-00 

21-36  J21-73 

22-12 

22-52 

22-92 

28-32  J24-12  124.90 

25-68 

DIMENSIONS    OF   THE    CKANK. 


297 


THICKNESS    OF    \VEB    OF   CRANK,  SUPPOSING   IT   TO    BE   CONTINUED 
TO    PADDLE-SHAFT    CENTRE. 


fe"s  j                                                    LENGTH   OF   STROKE    IN   FEET. 
HI 

S&"3     2 

2* 

3        3£ 

4 

4i 

5 

5£ 

6 

7 

8 

9 

20     2-64 

2-72     2-81     3-02 

3-2.3 

8-36 

8-49 

3-59 

8-70 

8-89 

4-06 

4-22 

21     2-75 

2-84     2-93     3-14 

3-34 

8-47 

8-61 

8-71 

3-82 

4-02 

4-19 

4-36 

22 

2-86 

2-96  ]  3-05     3-25     3'45 

3-58 

3-73 

8-S8 

8-94 

4-14 

4-32 

4-50 

23 

2-97 

8-08  i  8-17     8-36 

8-56 

8-69 

3-85 

8-95 

4-06 

4-26 

445 

4-64 

24 

8-08 

8-19 

8"29     8-47 

3-67 

8-80 

8-97 

4-07 

4-18 

4-38 

4-58 

4-77 

25 

8-19 

3-80 

8-41     3-58 

8-78 

8-91 

4-09 

4-19 

4-30 

4-50 

4-71 

4-90 

26 

3-30 

8-41 

8-52     8-69 

8-S9 

4-02 

4-20 

4-30 

4-42 

4-62 

4-84 

5-03 

27 

3-41 

8-52 

8-63     8-80 

4-00 

4'18 

4-31 

4'41 

4-53 

4-74 

4-97 

5-16 

28 

8-52 

8-63 

8-74 

8-91 

4-10 

4-24 

4-42 

4-52 

4-64 

4-86 

5-10 

5-29 

29 

8-63 

3-74 

3-85 

4-02 

4-20 

4-35 

4-53 

4-63 

4-75 

4-98 

5-22 

5-42 

SO 

8-75 

3-85 

8-96  !  4-18 

4-30 

4-46 

4-62 

4-74 

4-87 

5-11 

5-34 

5-55 

81 

8-38 

8-94 

4-06  1  4-28 

4-89 

4-56 

4-72 

4-85 

4-98 

5-23 

5-46 

5-67 

32 

8-91 

4-03 

4-16 

4-33 

4-49 

4-66 

4-82 

4-96 

5-09 

5-84 

5-57 

5-79 

83 

8-99 

4-12 

4-26 

4-48 

4-59 

4-76 

5-92 

5-07 

6-20 

5-45 

5-69 

5-91 

34 

4-07 

4-21 

4-36 

4-53 

4-69 

4-86 

5-02 

618 

5-31 

657 

5-80 

6-03 

35 

4-15 

4-30 

4-45 

4-62 

4-79 

4-96 

5-12 

5-28 

5-42 

5-68 

5-92 

6-15 

86 

4-22 

4-88 

4-54 

4-71 

4'89 

5-06 

P-22 

6'38 

5-53 

5-79 

6-03 

6-27 

87 

4-29 

4-46 

4-63 

4-80 

4-98 

5-16 

5-32 

5-48 

5-64 

5-90 

6-15 

6-39 

83 

4-86 

4-54 

4-72 

4-89 

5-08 

5-26 

6-42 

6-58 

5-74 

6-01 

6-26 

6-51 

39 

4-43 

4-62 

4-81 

4-.»S 

6-18 

5-36 

5-52 

5-68 

6-84 

6-12 

6-88 

6-62 

40 

4-50 

4-70 

4-90 

5-09 

5-28 

5-45 

5-62 

6'78 

5-94 

6-23 

6-49 

6-74 

41 

4-60 

4-80 

6-00 

6-19 

5'88 

5-54 

5-70 

5-86 

6-08 

6-34 

6-61 

6-86 

42 

4-70 

4-90 

5-10 

5-29 

5'48 

5-63 

5-78 

5-94 

6-11 

6-45 

6-78 

6-97 

43 

4-80 

6-00 

5-20 

5-89 

5'58 

5-72 

6-86 

6-02 

6-20 

6-56 

6-85 

7-08 

44 

4-89 

5-09 

5-30 

5-49 

6-68 

5-81 

5-94 

610 

6-28 

6-67 

6-97 

7-19 

45 

4-98 

5.18 

5-39 

6-58 

5'78 

5-90 

6-02 

6-18 

6-37 

6-77 

7-09 

7-30 

46 

6-07 

6-27 

5-48 

5-67 

5-87 

5-98 

6-10 

6'26 

6-45 

6-87 

7-21 

7-41 

46 

6-25 

5-45 

6-66 

5-85 

6-05 

6-14 

6-26 

6'42 

6-62 

7-07 

7-45 

7-63 

50 

5-43 

6-68 

6-84 

6-08 

6-23 

6-30 

6-38 

6-58 

6-79 

7-27 

7-67 

7-85 

52 

6-61 

5-81 

6-03 

6-28 

6-48 

6'52 

6-62 

6-81 

7-08 

7-47 

7-87 

8-07 

54 

6-78 

5-98 

6"21 

6-42 

6-68 

6-74 

6-86 

7-04 

7-27 

7-67 

8-05 

8-29 

56 

5-94 

6-14 

6-39 

6-60 

6'82 

6-96 

7-10 

7-27 

7-50 

7-87 

8-23 

8-51 

58 

6-10 

6-30 

6-57 

6-78 

7-00 

7-16 

7-88 

7'50 

7-72 

8-07 

8-41 

8-71 

60 

6-26 

6-50 

6-75 

6-96 

7-18 

7-86 

7-55 

7*78 

7-92 

8-27 

8-59 

8-91 

62 

6-43 

6-71 

6-95 

7-16 

7-86 

7-56 

7-75 

7-98 

8-11 

8-47 

8-79 

9-11 

64 

6-70 

6-91 

7-15 

7-84 

7-54 

7-74 

7-94 

8-18 

8-30 

8-65 

8-97 

9-31 

66 

6-92 

7-10 

7-33 

7-52 

7-72 

7-92 

8-12 

8-31 

8-49 

8-84 

9-17 

9-52 

68 

7-12 

7-80 

751 

7-70 

7-90 

8-10 

8-80 

8-49 

8-68 

9-04 

9-87 

9-72 

70 

7-32 

7-50 

7-69 

7-88 

8-08 

8-28 

8-48 

8-67 

8-87 

9-24 

9-59 

9-92 

72 

7-51 

7-69 

7-87 

8-07 

8-28 

8-46 

8-66 

8-85 

9-07 

9-48 

9-88 

10-12 

74 

7-70 

7-83 

8-06 

8-26 

8-47 

8'64 

8-82 

9-08 

9-26 

9-71 

10-06 

10-82 

76 

7-89 

8-07 

8-25 

8-46 

8-67 

8-82 

8-98 

8-21 

9-44 

9-98 

10-28 

10-53 

78 

8-08 

8-26 

8-44 

8-65 

8-87 

9-00 

9-14 

9-89 

9-62 

10-16 

10-51    10-78 

80 

8-27 

8-44 

8-62 

8-84 

9-06 

8-18 

9-30 

9-55 

9-80    10-87 

10-75 

10-94 

82 

8-46 

Ml 

8-81 

9-02 

9-24 

9-87 

9-41 

9-75 

9-99    10-58 

10-91 

11-13 

84 

8-65 

8-82 

9-00 

9-20 

9-42 

8-56 

9-72 

9-94    10-18  ;  10-69 

11-07 

11-31 

86 

8-85 

9-01 

8-18 

8-88 

9-60 

8-74 

9-92    10-18    10.86    10'84 

11.28 

11-50 

88 

9-04 

9-20 

9-36 

8-56 

9-75 

8-88 

10-18    10-38    10-54    10-98 

11-37    11-69 

90 

9-28 

'.>"•?.> 

9-55 

9-74 

9-98 

10-18  110-88  i  10-58    10-78    11-18 

11-51    11-89 

92 

8-48 

'.>•:>-, 

9-74 

8-98 

10-11    10-82    10-51    10-71    10-92    11-32  :ll-69    12-OS 

94 

8-68 

9-78 

9-93 

10-12 

10-80    10-50  J10-70    10-90    11-11    11-50    11-89 

12-27 

96 

8-82 

9-97    10-12 

10-80 

10-49    10-69    10-89    11-09    11-29  ;  11-68    12-08 

12-46 

98 

10-01  S  10-16    10-81 

10-49 

10-68 

10-88    11-08    11-27    11-48  111-87 

12-27 

12-64 

100    10-20 

10-35 

10-50 

10-68 

10-86 

11-06    11-26    11-46    11-66 

12-06 

12-45 

12-84 

13* 


298 


PBOPOETIONS    OF   STEAM-ENGINES. 


THICKNESS    OF   LARGE    EYE    OF    CRANK. 


•s-a 
*•  »  « 
f  "i-s 

e-.g 

5° 

LENGTH   OF   STROKE   IN   FEET. 

2 

H 

3 

»i 

4 

4* 

5 

Bi 

6 

7 

8 

9 

20 

1-71 

1-80 

1-89 

1-97 

2-05 

2-12 

219 

2-25 

2-32 

2-44 

2-55 

2-64 

21 

1-77 

1-87 

1-95 

2-04 

2-13 

2-20 

2-27 

2-32 

2-40 

2-52 

2-64 

2-75 

22 

1-83 

1-93 

2-01 

2-11 

2-20 

2-28 

2-85 

2-39 

2-48 

2-60 

2-72 

2-86 

23 

1-89 

1-99 

2-07 

2-18 

2-28 

2-36 

2-43 

2-46 

2-56 

2-68 

2-80 

2-97 

24 

1-95 

2-06 

2-14 

2-25 

2-35 

2-44 

2-51 

2-54 

2-64 

2-76 

2-88 

3-08 

25 

2-01 

2-12 

2-21 

2-32 

2-43 

2-52 

2-59 

2-62 

272 

2-84 

2-96 

3-18 

26 

2-07 

2-19 

2-28 

2-39 

2-50 

2-59 

266 

2-70 

2-80 

2-92 

3-04 

3-28 

27 

2-18 

2-25 

2-35 

2-46 

2-58 

2-66 

2-73 

2-78 

2-87 

2-99 

812 

3-38 

28 

2-19 

2-31 

2-42 

2-53 

2-65 

2-78 

2-80 

2-86 

2-94 

8-06 

8-20 

3-43 

29 

2-25 

2-37 

2-49 

2-60 

2-73 

2-80 

2-87 

2-94 

8-01 

813 

8-28 

3-58 

80 

2-30 

2-48 

2-56 

2-68 

2-80 

2-87 

2-94 

8-01 

8-08 

818 

3-36 

3-68 

31 

2-36 

2-50 

2-63 

2-74 

2-87 

2-94 

8-00 

3-07 

8-15 

8-26 

3-43 

3-73 

82 

2-42 

2-56 

2-69 

2-80 

2-94 

8-01 

3-06 

813 

8-22 

8-84 

8-50 

3-78 

83 

2-49 

2-62 

2-75 

2-86 

3-00 

8-08 

312 

8-20 

8-29 

8-41 

8-57 

8-83 

84 

2-55 

2-69 

2-81 

2-92 

3-06 

815 

313 

8-27 

3-36 

8-48 

3-64 

3-88 

85 

2'61 

2-75 

2-87 

2-98 

3-12 

8-22 

8-25 

8-34 

8-48 

8-55 

8-71 

4-93 

86 

2-67 

2-81 

2-93 

8-04 

8-18 

8-29 

832 

3-41 

3-50 

8-62 

3'78 

4-98 

87 

2-74 

2-87 

2-99 

8-10 

8-24 

8-85 

889 

8-48 

8-57 

8-69 

8-85 

4-03 

88 

2-81 

2-94 

3-05 

3-16 

8-30 

8-41 

846 

3-55 

8-64 

3-77 

8-91 

4-07 

39 

2-87 

8-00 

8-11 

3-22 

8'36 

8-47 

8-53 

3-62 

3-71 

3-84 

8-97 

411 

40 

2-93 

8-05 

8-17 

3-29 

3-42 

3-51 

8-60 

3'69 

8-78 

8-90 

4-03 

415 

41 

8-08 

8-13 

8-24 

8-37 

8-49 

8-58 

8-67 

3-75 

3-85 

3-98 

411 

4-24 

42 

3-13 

8-22 

3-31 

3-45 

8-56 

3-65 

8-74 

8'81 

3-92 

4-06 

419 

4-82 

43 

8-23 

8-80 

8-39 

3-53 

8-63 

3-72 

3-81 

8'87 

8-99 

414 

4-27 

4-40 

44 

8-83 

8-39 

8-47 

8-60 

8-69 

3-79 

8-88 

8-94 

4-06 

4-22 

4-85 

4-48 

45 

8-43 

347 

8-55 

367 

8-75 

8-86 

8-95 

4-01 

413 

4-30 

4-43 

4-56 

46 

8-53 

8-56 

8-68 

8-74 

8-81 

3'93 

4-02 

4-08 

4-19 

4-88 

4-51 

4-64 

48 

8-73 

3-73 

8-79 

8-88 

3-93 

4'06 

415 

4-22 

4-32 

4-52 

4'66 

4-80 

50 

3-93 

3-92 

3-95 

4-02 

4-09 

4-18 

4-27 

4-36 

4-46 

4-64 

4-81 

4-96 

52 

4-10 

4-05 

4-09 

4-18 

4-24 

4-31 

4-41 

4-48 

4-60 

4-78 

4-95 

5-10 

54 

4-28 

4-19 

4-24 

4-33 

4-38 

4-45 

4-55 

4-61 

4-74 

4-92 

5-09 

5-24 

56 

4-45 

4-83 

4-40 

4-47 

4-52 

4-59 

4-69 

4-75 

4-88 

5-06 

5-23 

5-38 

58 

4-62 

4-46 

4-56 

4-61 

4-66 

4-73 

4-83 

4-89 

5-00 

5-19 

5-37 

552 

60 

4-79 

4-60 

4-72 

4-76 

4-80 

4-87 

4-95 

5-03 

512 

5-21 

5-49 

5-66 

62 

4-98 

4-80 

4-88 

4-90 

4-96 

5-02 

5-09 

517 

5-26 

5-45 

5-63 

5-80 

64 

5-16 

5-00 

5-04 

5-03 

510 

516 

5-23 

5-31 

5-40 

5-59 

5-77 

5-94 

66 

5-84 

5-20 

5-20 

6-15 

5-24 

5-30 

5-37 

5-45 

5-54 

5-73 

5-90 

6-08 

68 

5-52 

5-40 

5-85 

5-27 

588 

5-44 

5-51 

5-59 

5-68 

5-87 

6-04 

6-22 

70 

5-70 

5-60 

5'49 

5-40 

5-52 

5-58 

5-64 

5-72 

5-80 

5-98 

616 

6-35 

72 

5-90 

5-78 

5-67 

5-56 

5-68 

5-74 

5-80 

5-86 

5-94 

612 

6-80 

6-49 

74 

6-09 

6-96 

5-84 

5-72 

584 

589 

5-95 

6-00 

6-08 

6-26 

6'44 

6-63 

76 

6-27 

6-14 

6-00 

5-88 

6-98 

6-03 

6-09 

614 

6-22 

6-40 

6-58 

6-75 

78 

6-46 

6-32 

6-16 

6-02 

6-12 

617 

6-23 

6-28 

636 

6-53 

6-71 

6-87 

80 

6-66 

6-49 

6-82 

6-16 

6-26 

6-31 

6-87 

6-43 

6-60 

6-66 

6-84 

7-02 

82 

6-86 

6-69 

6-50 

6-35 

6-42 

6-46 

6-51 

6-57 

6-64 

6-80 

6-98 

716 

84 

7-06 

6-88 

6-68 

6-54 

6-58 

6-61 

6-65 

6-71 

6-78 

6-94 

712 

7-30 

86 

7-27 

7-06 

6-84 

6-78 

6-74 

6-76 

6-79 

6-85 

6'90 

7-08 

7'2fi 

7-44 

88 

7-47 

7'24 

7'00 

6-92 

6-90 

6-91 

6-93 

6.99 

7'02 

7-21 

7-39 

7-57 

90 

7-67 

7-42 

718 

7-11 

7-05 

7-06 

7-08 

711 

714 

7-34 

7-51 

7-69 

92 

7-88 

7-62 

7-37 

7-29 

7-21 

7-22 

7-28 

7-27 

7-80 

7-48 

7-65 

7-83 

94 

8-09 

7-81 

7-55 

7-47 

7-87 

7-88 

7-88 

7-43 

7-45 

7-62 

7'79 

7-97 

96 

8-31 

7-99 

7-78 

7-65 

7-58 

7-54 

7-58 

7-59 

7-60 

7-76 

7'93 

811 

98 

8-52 

8-17 

7-91 

7-81 

7-69 

7-70 

7'68 

7-78 

7-76 

790 

8-06 

8-24 

100 

8-72 

8-41) 

•8-09 

7-97 

7-86 

7-84 

7-83 

787 

7-91 

8-04 

819 

8-30 

DIMENSIONS   OF   AIR-PUMP   AND   PISTON  ROD.      299 


1 

1 

DIMENSIONS  OP  THE  SEVERAL  PARTS   OF  PISTON   ROD  IN  INCHES. 

!!< 

->. 

o 

I 

i 

5 

PH     . 

3      1  43        S  « 

-M       ,            8  *»       .          "SfU 

u 

|| 

II  • 

id 

3§ 

1 

;jfl 

11 

a 

'$ 

•~  o 

8  a 
**Js   , 
^3  3 

«  ji  a     H  a 

S&-J    S'Z 

51 

5    3 
°31 

5-1 

§1 

igd 

laS" 

H 

:  — 

:  a 

£S 

6  °  £  '  Jf-  S-  a 

!|| 

go  g 

J3"g 

-5?  s 

I  J  °- 

J-S 

a  2 

5 

3 

5 

>3 

S           5 

S 

a 

* 

0*" 

E"1 

3 

20 

12-0 

2-0 

4-0 

1-90 

1-80     2-80 

2-30 

2-11 

•42 

1-70 

•70 

1-34 

21 

12-6 

2-1 

4-2 

1-99 

1-89     2-94 

2-41 

2-21 

•44 

1-78 

•73 

1-40 

22 

13-2 

2-2 

4-4 

2-09 

1-98     3-08 

2-53 

2-32 

•46 

1-87 

•77 

1-47 

23 

8-8 

2-3 

4-6 

2-18 

2-07     3-22 

2-64 

2-42 

•48 

1-95 

•80 

1-53 

24 

14-4 

2-4 

4-8 

2-28 

2-16 

8-36 

2-76 

2-53 

•50 

2-04 

•84 

1-60 

25 

15-0 

2-5 

5-0 

2-37 

2-25 

8-50 

2-87 

2-63 

•52 

212 

•87 

1-67 

26 

15-6 

2-6 

5-2 

2-47 

2-34 

8-64 

299 

2-74 

•54 

2-21 

•90 

1-73 

27 

16-2 

2-7 

5-4 

2-56 

2-43 

8-78 

3-11 

2-84 

•57 

2-29 

•94 

1-80 

28 

16-8 

2-8 

5-6 

2-66 

2-52 

8-92 

8-22 

2-95 

•59 

2-38 

•97 

1-87 

29 

17-4 

2-9 

5-8 

2-75 

2-61 

4-06 

M  ;U 

3-05 

•61 

2-46 

1-00 

1-94 

80 

18-0 

8-0 

6-0 

2-85 

2-70 

4-20 

3-45 

8-16 

•63 

2-55 

1-04 

2-00 

81 

18-6 

8-1 

6-2 

2-94 

2-79 

4-84 

8-57 

8-26 

•65 

2-03 

1-07 

2-07 

82 

19-2 

8-2 

6-4 

8-04 

2-S8 

443 

8-68 

8-87 

•67 

2-72 

1-10 

2-14 

83 

19-8 

8-3 

6-6 

8-18 

2-97 

4-62 

8-80 

8-47 

•69 

2-80 

1-14 

2-21 

84 

20-4 

8-4 

6-8 

8-23 

8-06 

4-76 

8-91 

8-57 

•71 

2-89 

1-19 

2-27 

85 

21-0 

8-5 

7-0 

8-32 

8-15 

4-90 

4-02 

8-67 

•73 

2-97 

1-22 

2-33 

86 

21-6 

8-6 

7-2 

8-42 

8-24 

6-04 

414 

8-78 

•75 

8-06 

1-26 

2-40 

87 

22-2 

8-7 

7-4 

8-51 

8-33 

518 

425 

8-88 

•78 

3-14 

1-29 

2-47 

88 

22-8 

8-8 

7-6 

8-61 

8-42 

5-32 

4-36 

8-99 

•80 

3-23 

1-33 

2-54 

89 

23-4 

8-9 

7-8 

8-70 

8'51 

5-46 

4-48 

4-09 

•82 

8-31 

1-86 

2-60 

40 

24-0 

4-0 

8-0 

8-80 

8-60 

5-60 

4-59 

4-20 

•84 

8-40 

1-40 

2-67 

41 

24-6 

4-1 

8-2 

8-89 

8-69 

5-74 

4-70 

4-80 

•86 

8-48 

1-43 

2-74 

42 

25-2 

4-2 

8-4 

8-99 

8-78 

6-88 

4-82 

441 

•89 

8-57 

1-47 

2-81 

48 

25-8 

4-3 

8-6 

4-08 

8-87 

6-02 

4-98 

4-51 

•91 

8-65 

1-50 

2-87 

44 

26-4 

4-4 

8-8 

4-18 

8-96 

6-16 

5-05 

4-62 

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8-74 

1-54 

2-98 

45 

27-0 

4-5 

9-0 

4-27 

4-05 

6-80 

5-17 

4-72 

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8-82 

1-57 

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27-6 

4-6 

9-2 

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414 

6-44 

5-28 

4-83 

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8-91 

1-61 

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28-8 

4-8 

9-6 

4-56 

4-82 

6-72 

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5-04 

1-02 

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1-68 

8-20 

50 

80-0 

5-0 

10-0 

4-75 

4-50 

7-00 

5-74 

6-25 

1-07 

4-25 

1-75 

8-33 

52 

81-2 

6-2 

104 

4-94 

4-68 

7-28 

5-97 

6-46 

1-11 

4-42 

1-82 

8-47 

64 

82-4 

5-4 

10-8 

5-13 

4-86 

7-56 

6-21 

667 

1-15 

4-69 

1-89 

8-60 

56 

88-6 

5-6 

11-2 

6-32 

5-04 

7-84 

6-44 

5-88 

1-19 

4-77 

1-96 

8-74 

58 

84-8 

5-8 

11-6 

5-51 

5-22 

8-12 

6-67 

6-09 

1-23 

4-94 

2-03 

8-88 

60 

86-0 

6-0 

12-0 

5-70 

5-40 

8-40 

6-90 

6-30 

1-27 

5-11 

2-10 

401 

62 

87-2 

6-2 

12-4 

6-69 

5-58 

8-68 

7-18 

6-51 

1-81 

5-28 

2-17 

4-14 

64 

88-4 

64 

12-8 

6-08 

576 

8-96 

7-36 

6-72 

1-85 

6-45 

2-24 

4-27 

66 

89-6 

6-6 

18-2 

6-27 

594 

9-24 

7-59 

6-98 

1-89 

5-62 

2-81 

4-40 

68 

40-8 

6-8 

18-6 

6-46 

612 

9-52 

7-82 

7-14 

1-40 

5-79 

288 

4-53 

70 

42-0 

7-0 

14-0 

6-65 

6-80 

9-80 

8-05 

7-85 

1-47 

5-96 

2-44 

4-67 

72 

43-2 

7-2 

14-4 

6-84 

6-48 

10-08 

8-28 

7-56 

1-51 

6-13 

2-51 

4-80 

74 

44-4 

7-4 

14-8 

7-03 

666 

10-86 

8-51 

7-77 

1-55 

6-80 

2-58 

4-93 

76 

45-6 

7-6 

15-2 

7-22 

6-84  |10  64 

8-74 

798 

1-67 

6-46 

2-66 

5-07 

78 

46-8 

7-8 

15-6 

7-41 

7-02  11092 

8-97 

8-19 

1-70 

6-63 

2-73 

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80 

48-0 

8-0 

16-0 

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7-20 

11-20 

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8-40 

1-73 

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82 

49-2 

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7-81 

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62-8 

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17-6 

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7-92  12-82 

10-12 

9-24 

1-85 

7-48 

8'08 

5-87 

90 

54-0 

9-0 

180 

8-66 

8-10  12-60 

10-84 

9-45 

1-89 

7-66 

8-15 

6-00 

92 

55-2 

9-2 

18-4 

8-75 

8-28  12-88 

10-57 

9-66 

1-92 

7-88 

8-22 

6-14 

94 

56-4 

9-4 

18-8 

8-94 

846   18-16 

10-80 

9-87 

195 

8-00 

8-29 

6-27 

96 

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9-6 

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8-64   13-44 

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19-6 

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8-82  18-72 

11-26 

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100 

60- 

10-0 

20O 

9-50 

9-00  14-00 

11-49 

10-50 

2.11 

8-51 

8-50 

6-66 

300 


PROPORTIONS    OF    STEAM-ENGINES. 


1 

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9-50 

116 

6-94 

2-32 

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82 

4-54 

5-12 

2-02 

5-99 

3-52     5-12 

65 

10-06 

125 

729 

2-36 

7-39 

33 

4-68 

5-28 

2-08 

6-18 

3-63 

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70 

10-56 

134 

7-63 

240 

7-55 

84 

4-83 

5-44 

2-14 

6-37 

3-74 

5-44 

75 

10-96 

143 

7-98 

2-44 

7-71 

85 

4-97 

5-60 

2-21 

6-56 

8-85 

5-60 

80 

11-31 

152 

8-32     2-48 

7'87 

36 

5-11 

5-76 

2-27 

6-74 

3-96 

5-76 

85 

11-61 

161 

8-67 

2-52 

8-03 

37 

5-26 

5-92 

2-23 

6-93 

4-07 

5-92 

90 

11-89 

170 

9-01 

2-56 

8-19 

88 

5-40 

6-08 

2-39 

7-12 

4-18 

6-08 

95 

12-09 

179 

9-36 

2-60 

S'85 

39 

5-54 

6-24 

2-45 

7-31 

4-29 

6-24 

100 

12-19 

188 

9-70 

2-64 

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40 

5-69 

6-40 

2-52 

7-50 

4-40 

6-40 

105 

12-29 

197 

10-05 

268 

8'66 

41 

5-88 

6-52 

2-58 

7-68 

4-51 

6-56 

110 

12-56 

206 

10-39 

2-72 

8-80 

42 

5-97 

6-63 

2-64 

7-87 

4-62 

6-72 

115 

12-83 

215 

10-74 

2-76 

8'94 

43 

6-11 

6-84 

2-71 

8-05 

4-73 

6-S8 

120 

13-10 

224 

1108 

2-80 

9'08 

44 

6-25 

7-00 

2-78 

8-24 

4-84 

7-04 

125 

13-37 

283 

11-43 

2-84 

9'22 

45 

6-39 

7-16 

2-84 

8-42 

4-95 

7-20 

130 

13-64 

242 

11-77 

2-88 

9-36 

46 

6-54 

7-32 

2-91 

8-61 

5-06 

7-36 

335 

18-91 

251 

12-12 

2-91 

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48 

6-82 

7-63 

3-03 

8-98 

5-28 

7-68 

140 

14-18 

260 

12-46 

2-94 

9-63 

50 

7-11 

7-95 

3-16 

9-35 

5-50 

8-00 

150 

14-70 

278 

13-15 

800 

9-89 

52 

7-39 

8-27 

328 

9-72 

5-72 

8-32 

160 

15-20 

296 

13-84 

3-07 

10-12 

54 

7-67 

8-59 

3-41 

10-09 

5-94 

8-64 

170 

1565 

814 

14-53 

8-14 

10-36 

56 

7-95 

8-91 

3-58 

10-46 

616 

8-96 

180 

16-09 

332 

15-22 

3-20 

10-60 

68 

8-24 

9-23 

8-66 

10-84 

6-38 

9-28 

190 

16-53 

350 

15-91 

3-26 

1084 

60 

8-52 

9-54 

8-78 

11-20 

6-60 

9-60 

200 

16-97 

868 

16-60 

8-82 

11-06 

62 

8-80 

9-86 

8-91 

11-57 

6-82 

9-92 

210 

17-39 

886 

17-29 

3-38 

11-29 

64 

9-09 

10-18 

4-04 

11-94 

7-04 

10-24 

220 

17-79 

404 

17-98 

8-44 

11-51 

66 

9-37 

10-50 

4-17 

12-31 

7-26 

10-56 

230 

18-19 

422 

18-67 

3-50 

1173 

68 

9-65 

10-82 

4-29 

12-68 

7-48 

1088 

240 

18-58 

440 

19-36 

3-56 

11-95 

70 

994 

11-13 

4-42 

13-06 

7-70  -11-20 

250 

18-97 

456 

20-05 

3-61 

1215 

72 

10-22 

11-45 

4-54 

13-44 

7-92   11-52 

260 

19-34 

476 

20-74 

3-66 

1285 

74 

10-50 

11-77 

4-67 

18-81 

8-14   11-84 

270 

19-70 

494 

21-43 

3-72 

1-2-55 

76 

10-79 

12-08 

4-79 

14-19 

8-36   12-16 

280 

2006 

512 

22-12 

3-77 

12-75 

78 

11-07 

12-40 

4-91 

14-56 

8'58   12-48 

290 

20-42 

580 

22-81 

3-82 

12-95 

80 

11-36 

12-72 

5-03 

14-94 

8-80  '12-80 

800 

20-78 

548 

23-50 

8-88 

13-14 

82 

11-64 

13-04 

5-16 

15-32 

9-02   13-12 

810 

21-12 

566 

24-19 

8-ya 

13-33 

84 

11-93    18-85 

5-29 

15-69 

9-24   13-44 

820 

21-46 

584 

24-83 

3-98 

18-51 

86 

12-21    18-67 

5-41 

16-07 

9-46  13-76 

830 

21-80 

602 

25-57 

4-03 

1369 

88 

12-50    18-99 

5-54 

16-44 

9-68  1  14-03 

840    22-14 

620 

2626 

4-07 

13-87 

90 

12-79    14-30 

5-67 

16-82 

9-90   14-40 

850    22-46 

638 

26-95 

4-12 

14-05 

92 

13-07   14-62 

5-80 

17-20 

10-12 

14-72 

360    22-77 

656 

27-64 

4-16 

14-^3 

94 

13-35   14-94 

5-92 

17-58 

10-34 

15-04 

370    23-09 

674 

28-88 

4-21 

14-41 

96 

J8-64   15-26 

6-05 

17-95 

10-56 

15-36 

880    23-40 

692 

29-02 

4-26 

14v9 

93 

18-92   15-58 

6-17 

18-87 

10-78   15-68 

890    23-70 

710 

29-72 

4-31 

14-76 

100 

14-20   15-90 

6-30 

13-75 

11-00  ;16-00 

400 

24-00 

728 

80-41 

4-87 

14-92 

DIMENSIONS    PROPER   FOR   LOCOMOTIVES.  301 

I  may  here  repeat  that  the  diameter  of  cylinder  in  inches  is 
given  in  the  first  vertical  column,  beginning  at  20  inches  and 
ending  at  100  inches,  while  the  length  of  the  stroke  in  feet  is 
given  in  the  first  horizontal  column,  beginning  with  2  feet  and 
ending  with  9.  If,  therefore,  we  wish  to  find  the  dimension 
proper  for  any  given  engine,  of  which  we  must  know  the  diam- 
eter of  cylinder  and  length  of  stroke,  we  find  in  the  first  vertical 
column  the  given  diameter  in  inches,  and  in  the  first  horizontal 
column  the  given  length  of  stroke  in  feet ;  and  where  the  vertical 
column  under  the  given  stroke  intersects  the  horizontal  column 
opposite  the  given  diameter,  there  we  shall  find  the  required 
dimension.* 

LOCOMOTIVE  ENGINES. 

It  would  be  a  mere  waste  of  time  and  space  to  recapitulate 
rules  similar  to  the  foregoing  as  applicable  to  locomotive  en- 
gines, since  the  strengths  and  other  proportions  proper  for  loco- 
motives can  easily  be  deduced  by  taking  an  imaginary  low  pres- 
sure cylinder  of  twice  the  diameter  of  the  intended  locomotive 
cylinder,  and  therefore  of  four  times  the  area,  when  the  propor- 
tions will  become  at  once  applicable  to  the  locomotive  cylinder 
with  a  quadrupled  pressure,  or  100  Ibs.  on  the  square  inch.  In 
locomotive  engines  the  piston  rod  is  generally  made  ^th  of  the 
diameter  of  the  cylinder,  whereas  by  the  mode  of  determining 
the  proportions  that  is  here  suggested  it  would  be  £th.  But 
piston  rods  are  made  of  their  present  dimensions,  not  so  much 
to  bear  the  tension  produced  by  the  piston,  as  to  bear  the  com- 
pression when  they  act  as  a  pillar ;  and  properly  speaking  the 
proportionate  diameter  should  diminish  with  every  diminution 

*  For  screw  or  other  short-stroke  engines  working  at  a  high  speed,  the  strengths 
of  shafts  given  In  the  foregoing  tables  should  be  somewhat  increased,  and  the 
length  of  bearing  at  least  doubled.  In  some  recent  screw  engines  an  Irregular 
motion  of  the  engine  has  been  perceived,  owing  to  the  elasticity  of  the  shaft  For 

3    //D\  s      ^i    *    B~ 

such  engines  a  correspondent  suggests  the  formula  A/  (  =-  I  +  -r —  _  diam- 
eter of  journal  in  Inches ;  where  D  =  diameter  of  the  cylinder  In  inches  and  K  = 
radius  of  crank  in  inches. 


302  PROPORTIONS    OF    STEAM-ENGINES. 

in  the  length  of  the  stroke.  In  very  short  cylinders  a  proportion 
of  ^  of  the  diameter  of  the  cylinder  would  suffice  in  the  case 
of  low  pressure  engines,  which  answers  to  Hh  of  the  diameter 
in  locomotives  where  the  stroke  is  always  very  short.  But  in 
high  pressure  engines  of  any  considerable  dimensions,  carrying 
100  Ibs.  on  the  inch,  the  diameter  of  the  piston  rod  should  he 
•5-th  of  the  diameter,  answering  to  ^rtth  of  the  diameter  in  low 
pressure  engines  of  the  common  total  pressure  of  25  Ibs.  per 
square  inch. 


CHAPTER  Y. 

PROPORTIONS  OF  STEAM-BOILERS. 

IN  proportioning  boilers  two  main  requirements  have  to  be 
kept  in  view :  1st.  The  provision  of  a  sufficient  quantity  of  grate- 
bar  area  to  burn — with  the  intended  velocity  of  the  draught — 
the  quantity  of  coals  required  to  generate  the  necessary  quantity 
of  steam ;  and  2d.  The  provision  of  a  sufficient  quantity  of  heat- 
ing surface  in  the  boiler,  to  make  sure  that  the  heat  will  be  prop- 
erly absorbed  by  the  water,  and  that  no  wasteful  amount  of 
heat  shall  pass  up  the  chimney.  Even  the  quantity  of  heating 
surface,  however,  proper  to  be  supplied  for  the  evaporation  of  a 
given  quantity  of  water  in  the  hour  will  depend  to  some  extent 
upon  the  velocity  of  the  draught  through  the  furnace :  for  upon 
that  velocity  will  depend  the  intensity  of  the  heat  within  the 
furnace,  and  upon  the  intensity  of  the  heat  will  depend  the 
quantity  of  water  which  a  given  area  of  surface  can  evaporate. 
The  first  point  therefore  to  be  investigated  is  the  best  velocity 
of  the  draught,  and  the  circumstances  which  determine  that 
velocity.  Here,  too,  there  are  two  guiding  considerations.  The 
first  is,  that  if  the  velocity  of  the  draught  be  made  too  great,  the 
small  coals  or  cinders  will  be  drawn  up  into  the  chimney  and 
be  precipitated  as  sparks,  causing  in  many  cases  serious  annoy- 
ance. The  second  consideration  is,  that  the  temperature  of  the 
escaping  smoke  should  be  as  low  as  possible,  and  should  in  no 
case  exceed  600°.  While,  therefore,  it  is  desirable  in  land  and 
marine  boilers  to  have  a  rapid  draught  through  the  furnace — 
such  as  is  produced  in  locomotives  by  the  blast-pipe — in  order 
that  the  heat  maybe  sufficiently  intense  to  enable  a  small  amount 
of  surface  to  accomplish  the  required  evaporation,  it  is  at  the 


304  PROPORTIONS    OF    STEAM-BOILERS. 

same  time  inadmissible  to  have  such  a  rapid  draught  in  the 
chimney  as  will  suck  up  and  scatter  the  small  particles  of  the 
coal;  nor  is  it  desirable  that  the  velocity  of  the  air  passing 
through  the  grate-bars  should  be  so  great  as  to  lift  small  pieces 
of  coal  or  cinder  and  carry  them  into  the  flues.  No  furnace  has 
yet  been  constructed  which  reconciles  the  conditions  of  a  high 
temperature  with  a  moderate  velocity  of  the  entering  air :  but 
such  a  furnace  may  be  approximated  to  by  making  the  opening 
through  the  fire-bridge  very  small,  and  by  insuring  the  necessary 
flow  of  air  through  these  small  openings  by  the  application  of  a 
horizontal  steam-jet  at  each  opening;  as  by  this  arrangement  a 
high  temperature  may  be  kept  up  in  the  furnace,  at  the  same 
time  that  the  contraction  of  the  area  through  or  over  the  bridge 
will  not  so  much  impair  the  draught  as  to  prevent  the  requisite 
quantity  of  coal  from  being  burnt. 

The  exhaustion  which  a  chimney  produces  is  the  effect  of  the 
greater  rarity  of  the  column  of  air  within  the  chimney  than  that 
of  the  air  outside.  If  the  air  be  heated  until  it  is  expanded  to 
twice  its  volume,  then,  its  density  being  half  of  what  it  was 
before,  each  cubic  inch  of  the  hot  air  will  weigh  only  half  as 
much  as  a  cubic  inch  of  cold  air ;  and  if  the  hot  air  be  enclosed 
in  a  balloon,  it  will  ascend  in  the  cold  air  with  a  force  of  ascent 
equal  to  half  the  weight  of  the  balloon  full  of  cold  air.  As  water 
is  about  773  times  heavier  than  air  at  the  freezing-point,  it  will 
require  773  cubic  inches  of  air,  heated  until  they  expand  to  twice 
their  volume,  to  have  ascensional  force  sufficient  to  balance  a 
cubic  inch  of  water :  or  if  a  syphon-tube  be  formed  with  a  col- 
umn of  water  1  inch  high  in  one  leg,  it  will  require  a  column  of 
the  hot  air  1546  inches  (or  nearly  129  feet)  high,  in  the  other 
leg,  to  balance  the  column  of  water  1  inch  high.  In  other  words, 
a  chimney  heated  until  the  density  of  the  smoke  is  only  half  that 
of  the  air  entering  the  furnace,  and  which  will  be  the  case  at  a 
temperature  under  600°,  will,  if  129  feet  high,  produce  an  ex- 
haustion of  1  inch  of  water.  In  land  boilers  the  ordinary  ex- 
haustion or  suction  of  chimneys  is  such  as  would  support  a  col- 
umn of  from  1  to  2  inches  of  water.  But  in  steam-vessels  the 
height  of  the  chimney  is  limited,  and  the  deficient  height  has  to 


PBOPER  HEIGHTS  FOR  CHIMNEYS.         305 

be  made  up  for  by  an  increased  area.  In  practice,  tbe  diameter 
of  the  chimney  of  a  steam-vessel  is  usually  made  somewhat  less 
than  the  diameter  of  the  cylinder,  there  being  supposed  to  be 
one  chimney  and  two  cylinders,  with  the  piston  travelling  at  the 
speed  usual  in  paddle  vessels. 

Boulton  and  "Watt's  rule  for  proportioning  the  dimensions  of 
the  chimneys  of  their  land  engines  is  as  follows : — 

BOULTON  AND  WATTES  BULE  FOE  FIXING  THE  PEOPES  SECTIONAL 
AEEA  OF  A  CHIMNEY  OF  A  LAND  BOILEE  WHEN  ITS  HEIGHT 
IS  DETERMINED. 

RULE. — Multiply  the  number  of  pounds  of  coal  consumed  under 

'  the  toiler  per  hour  by  12,  and  divide  the  product  ~by  the  square 

root  of  the  height  of  the  chimney  in  feet:  the  quotient  is  the 

proper  area  of  the  chimney  in  square  inches  at  the  smallest 

part. 

Example. — "What  is  the  proper  sectional  area  of  a  factory 
chimney  80  feet  high,  and  with  a  consumption  of  coal  in  the 
furnace  of  300  Ibs.  per  hour? 

Here  300  x  12  =  3,600 ;  and  divided  by  9  (the  square  root 
of  the  height  nearly)  we  get  400,  which  is  the  proper  sectional 
area  of  the  chimney  in  square  inches.  If  therefore  the  chimney 
be  square,  it  will  measure  20  inches  each  way  within. 

BOTTLTON  AND  WATT'S  EtTLE  FOE  FIXING  THE  PEOPEE  HEIGHT  OF 
THE  CHIMNEY  OF  A  LAND  BOILEE  WHEN  ITS  SECTIONAL  AEEA 
IS  DETEEMINED. 

KULE. — Multiply  the  number  of  pounds  of  coal  consumed  under 
the  boiler  per  hour  by  12,  and  divide  the  product  by  the  sec- 
tional area  of  the  chimney  in  square  inches :  square  the  quo- 
tient thus  obtained,  which  will  give  the  proper  height  of  the 
chimney  in  feet. 

^Example.— What  is  the  proper  height  in  feet  of  the  chimney 
of  a  boiler  which  burns  300  Ibs.  of  coal  per  hour,  the  sectional 
area  of  the  chimney  being  400  square  inches  ? 

Here  300  x  12  =  3,600,  which  divided  by  400  (the  sectional 


306  PROPORTIONS    OF   STEAM-BOILERS. 

area)  =  9,  the  square  of  which  is  81 ;  and  this  is  the  proper 
height  of  the  chimney  in  feet. 

These  rules,  though  appropriate  for  land  boilers  of  moderate 
size,  are  not  applicable  to  powerful  boilers  with  internal  flues, 
such  as  those  used  in  steam-vessels,  in  which  the  sectional  area 
of  the  chimney  is  usually  adjusted  in  the  proportion  of  6  to  8 
square  inches  per  nominal  horse-power.  This  will  plainly  appear 
from  the  following  investigation  : — 

In  a  marine  boiler  suitable  for  a  pair  of  engines  of  110-horse- 
power,  the  area  of  the  chimney,  allowing  8  square  inches  per 
nominal  horse-power,  would  be  880  square  inches.  Supposing 
the  boiler  to  consume  10  Ibs.  of  coal  per  nominal  horse-power 
per  hour,  or  say  10  cwt.  (or  1120  Ibs.)  of  coal  per  hour,  and  that 
the  chimney  was  46  feet  high,  then,  by  Boulton  and  "Watt's  rule 
for  land  engines,  the  sectional  area  of  the  chimney  should  be 
1120  x  12  -,- V  46  =  13,440-?- say  7=1,920  square  inches.  This, 
it  will  be  observed,  is  more  than  twice  the  area  obtained  by 
allowing  a  sectional  area  of  8  square  inches  per  nominal  horse- 
power. Here,  therefore,  is  a  discrepancy  which  it  is  necessary 
to  get  to  the  bottom  of. 

In  Peclet's  '  Treatise  on  Heat '  an  investigation  is  given  of 
the  proper  dimensions  of  a  chimney,  which  investigation  is 
recapitulated  and  ably  expanded  by  Mr.  Rankine.  But  it  gives 
results  similar  to  those  deduced  from  Boulton  and  Watt's  rule 
for  their  small  land  boilers,  and  the  expressions  are  much  more 
complicated.  Thus  if  w  =  the  weight  of  fuel  burned  in  a  given 
furnace  per  second;  V0=the  volume  of  air  at  32°  required 
per  Ib.  of  fuel,  and  which  in  the  case  of  common  boilers  with  a 
chimney  draught  is  estimated  at  300  cubic  feet ;  Ti  =  the  abso- 
lute temperature  of  the  smoke  discharged  by  the  chimney,  and 
which  is  equal  to  the  temperature  shown  by  the  thermometer  + 
461 '2°;  T0=the  absolute  temperature  of  the  freezing-point,  or 
461 '2°+  32°;  A  =  the  sectional  area  of  the  chimney  in  square 
feet ;  and  u  =  the  velocity  of  the  current  in  the  chimney  in  feet 
per  second : 

M>Y0T. 

Then  u  = 


VELOCITY    OF   DRAUGHT    IN   CHIMNEYS.  307 

If  now  I  =  the  length  of  the  chimney  and  of  the  flue  leading 
to  it  in  feet ;  m  =  the  mean  hydraulic  depth  of  the  smoke,  or 
the  area  of  the  flue  divided  hy  its  perimeter,  and  which  for  a 
round  flue  and  chimney  is  J  of  the  diameter;  f=  a  coefficient 
of  friction,  the  value  of  which  for  a  current  of  gas  moving  over 
sooty  surfaces  Peclet  estimates  at  0'012 ;  G  a  factor  of  resistance 
for  the  passage  of  the  air  through  the  grate,  and  which  in  the 
case  of  furnaces  burning  20  to  24  Ibs.  of  coal  per  hour  on  each 
square  foot,  Peclet  found  to  be  12 ;  h  =  the  height  of  the  chim- 
ney in  feet :  Then  by  a  formula  of  Peclet's 


2. 

which  formula,  with  the  value  that  Peclet  assigns  to  the  con- 
stants, becomes 


and  by  transposition  and  reduction 


where  64£  is  twice  the  power  of  gravity,  or  32^. 

If  now  the  chimney  be  made  46  feet  high  and  the  flue  leading 
to  it  be  3  feet  diameter  and  64  feet  long,  then  64-3  x  46  = 
2957-8  ;  -012  x  100  =  1'2  ;  m  =  Jth  of  3,  or  f,  or  '75,  and  1-2  H- 
•75  =  1-6. 

Hence  the  equation  becomes 


But  ?/  —         ° 
-    AT0 

Hence  W-     °    '  =  14-23 


Now  if  1,120  Ibs.  of  coal  be  consumed  per  hour,  -31  Ibs. 
will  be  consumed  per  second  =  w ;  and  if  the  temperature  of 


808  PROPORTIONS    OF    STEAM-BOILERS. 

the    chimney  be    G00°,    then    600°  +  461°  =  1061°  =  T,,  and 
461°  +  32°  =  493°  =  T0. 

•31  x  300  x  1001 
Hence  ~    — 493  A —    ~~  =  14'6 

A  =  14  square  feet,  or  2,010  square  inches;  whereas  1,920 
square  inches  is  the  area  given  by  Boulton  and  Watt's  rule. 
Peclet's  rule,  consequently,  gives  areas  much  too  great  for  boilers 
with  internal  flues,  though  it  will  answer  pretty  well  for  small 
land  boilers  with  external  flues :  but  even  here  it  has  the  disad- 
vantage of  being  too  complicated  for  common  use.  It  is  clear 
that  the  friction  of  the  smoke  passing  through  internal  flues  must 
be  much  less  than  the  friction  of  smoke  passing  through  external 
flues  like  that  which  surrounds  a  wagon-boiler.  For  as  only 
one  side  of  the  external  flues  is  efficient  in  heating,  the  flue  with 
the  same  friction  per  foot  in  length  will  require  to  be  nearly 
three  times  as  long  as  in  the  case  of  an  internal  flue  of  the  same 
area,  to  give  the  required  amount  of  heating  surface.  In  steam 
vessels  much  heat  is  wasted,  from  the  height  of  the  chimney 
being  necessarily  so  limited  that  but  a  small  portion  of  the  as- 
censional force  due  to  the  temperature  of  the  smoke  is  obtained. 
Thus,  if  a  height  of  chimney  of  129  feet  will  produce  an  exhaus- 
tion of  an  inch  of  water  when  the  heat  is  sufficient  to  expand 
the  air  into  twice  its  volume,  as  will  be  the  case  at  a  tempera- 
ture considerably  under  000°,  then  it  is  clear  that  another  height 
of  129  feet,  added  to  the  first,  would  produce  an  exhaustion 
equal  to  a  column  of  two  inches  of  water  without  any  additional 
expenditure  of  heat ;  and  this  increase  would  go  on  until  the 
velocity  of  the  draught  became  such  that  the  friction  of  the  ad- 
ditional height  balanced  its  ascensional  force.  In  steam-vessels, 
where  the  chimney  is  necessarily  short,  a  great  part  of  the  ex- 
hausting or  rarefying  effect  of  the  heat  is  lost ;  and  in  steam- 
vessels,  therefore,  a  chimney-draught  is  a  more  wasteful  expe- 
dient for  promoting  combustion  than  it  is  in  the  case  of  a  land 
boiler,  where  a  much  larger  proportion  of  the  ascensional  power 
of  the  heat  may  be  made  available. 

The  proportion  of  heating  surface  per  nominal  horse-power 


PROPER  AREA  OF  HEATING  SURFACE.       309 

obtaining  in  marine  boilers  varies  very  much  in  different  exam- 
ples, being  in  some  boilers  12  square  feet,  in  others  17  square 
feet,  in  others  20  square  feet,  in  others  30  square  feet,  and  in 
some  as  much  as  35  square  feet  per  nominal  horse  power.  In 
fact,  the  proportion  of  heating  surface  required  will  depend  upon 
the  intended  ratio  in  which  the  nominal  is  to  exceed  the  actual 
power,  which  is  now  often  as  much  as  8  or  9  times,  and  also 
upon  the  measure  of  expansive  action  which  is  proposed  to  be 
adopted.  In  marine  boilers,  as  in  land  boilers,  about  9  square 
feet,  or  1  square  yard,  of  heating  surface  will  be  required  to  boil 
off  a  cubic  foot  of  water  in  the  hour,  and  in  Boulton  and  Watt's 
modern  marine  tubular-boilers  they  allow  10  square  feet  of  heat- 
ing surface  to  evaporate  a  cubic  foot  of  water  in  the  hour,  10 
square  inches  of  sectional  area  of  tubes,  7  square  inches  of  sec- 
tional area  of  chimney,  and  14  square  inches  of  area  over  the 
furnace  bridges.  The  proportions  of  modern  flue-boilers  are  not 
very  different,  except  that  there  is  greater  sectional  area  of  flue. 
But  no  attempt  has  yet  been  made  to  connect  the  proportions 
proper  SOT  small  land  boilers,  with  those  proper  for  large  marine 
boilers,  or  to  construct  a  rule  that  would  be  applicable  to  every 
class  of  flue-boilers. 

Great  confusion  has  been  caused  by  referring  to  so  indefinite 
a  unit  as  the  nominal  power  of  a  boiler,  and  it  is  much  bet- 
ter to  make  the  number  of  cubic  feet  which  the  boiler  can 
evaporate  the  measure  of  its  power.  This  again  depends  upon 
the  intensity  of  the  draught.  But  it  may  be  reckoned  that  5  or 
6  square  feet  of  surface  will  evaporate  a  cubic  foot  per  hour  in 
locomotive  boilers,  and  9  or  10  square  feet  in  land  and  marine 
boilers. 

The  main  dimensions  and  proportions  of  Boulton  and  Watt's 
wagon-boilers  of  different  powers  are  given  in  the  following 
table : — 


310 


PROPORTIONS    OF    STEAM-BOILERS. 


PROPORTION   OP  BOITLTON  AND   WATT'S   WAGON  BOILERS. 


Power. 

Length 
Boiler. 

Breadth 
BoHer. 

Depth 
Boiler. 

Mean 
Height 
of  Flue. 

Breadth 
Fl°ue. 

Sectional 
Area 
of   Flue. 

Sectional 
Area 
of  Flue 
per  H.  P. 

ft.    in. 

ft    in. 

ft.    in. 

in. 

in. 

sq.  in. 

sq.  in. 

2 

4    0 

3        2 

4        1 

20 

9 

ISO 

90 

8 

5    3 

3       4 

4        4 

21 

9 

189 

63 

4 

6    0 

3        6 

4       7 

22 

10 

220 

55 

6 

7    0 

3       9 

5        1* 

27 

10 

270 

45 

8 

8    0 

4       0 

5        6 

81 

12 

872 

44 

10 

9    0 

4       8 

5        91 

35 

12 

400 

40 

12 

10    0 

4        6 

6        0 

36 

13 

468 

39 

14 

10    0 

4       9 

6       21 

39 

13 

507 

86 

16 

11    9 

5       0 

6        6 

40 

14 

560 

85 

18 

12    8 

5       2 

6        8 

42 

14 

588 

82 

20 

13    6 

5       4 

6      11 

44 

H 

616 

80 

80 

16    0 

5        6 

7        3 

45 

15 

720 

24 

45 

19    0 

6        0 

8       5 

53 

16 

795 

17 

These  proportions  enable  us  to  establish  the  following  rule, 
which  is  applicable  to  flue-boilers  of  every  class : — 

TO    DETERMINE    THE    PROPER    SECTIONAL   AREA   OF  THE    FLUE  IN 
FLTJE-BOILERS. 

ECXE. — Multiply  the  square  root  of  the  number  of  pounds  of 
coal  consumed,  per  hour  fiy  the  constant  number  300,  and  di- 
vide the  product  ly  the  square  root  of  the  height  of  the  chim- 
ney in  feet :  the  quotient  is  the  proper  sectional  area  of  the 
flue  in  square  inches. 

Example  1. — "What  is  the  proper  sectional  area  of  the  flue  in 
a  flue-boiler  burning  100  Ibs.  of  coal  per  hour,  the  chimney  being 
49  feet  high. 

Here  V  100  =  10,  and  10  x  300  =  3000;  which  divided  by 
7  (the  square  root  of  49)  =  428  square  inches,  which  is  the 
proper  area  of  the  flue  in  this  boiler. 

Example  2. — What  is  the  proper  sectional  area  of  the  flue  in 
a  flue-boiler  burning  30  Ibs.  of  coal  per  hour,  the  chimney  being 
81  feet  high  ? 

Here  V30  =  5 '48,  and  5-48  x  300  =  1644;  which  divided 
by  9  (the  square  root  of  81)  =  183  nearly,  which  is  the  proper 
area  of  the  flue  in  square  inches. 


BOULTON  AND  WATT'S  PRACTICE. 


311 


Example  3. — "What  is  the  proper  area  of  the  flue  in  a  flue- 
boiler  burning  1,000  Ibs.  of  coal  per  hour,  and  with  the  chimney 
49  feet  high  ? 

Here  VlOOO  =  31'78,  which  x  300  =  9534,  and  dividing  by 
V  (which  is  the  square  root  of  49),  we  get  1,362,  as  the  proper 
area  of  the  flue  in  square  inches.  This  is  equivalent  to  13 '62 
square  inches  per  horse- power. 

It  is  the  universal  experience  with  boilers  of  every  class,  that 
large  boilers  are  more  economical  than  small,  or,  in  other  words, 
that  a  given  quantity  of  coal  will  boil  off  more  water  in  boilers 
of  large  power  than  in  boilers  of  small  power.  Nevertheless, 
for  purposes  of  classification,  it  may  be  convenient  to  assume 
the  efficiencies  as  equal. 

The  proper  proportions  of  flue-boilers  from  1  to  100  horses 
power  are  given  in  the  following  Table : — 

PEOPEE  PBOPOBTIOXS  OF  FLTJE-BOILEBS  OF  DIFFEEESTT  POWEE3. 


Hone  Power. 

Pounds  of 
Coal 
contained 
per  boar. 

Sectional 
Area  of  Flue 
in  B.  &  W.'t 

boilers. 

Sectional 
Area  of  Flue 
by  rule, 
with  chim- 
ney 49  feet 
high. 

Sectional 
Area  of  Flue 
by  rule, 
with  chim- 
ney 81  feet 
high 

Heating 
per"lj!  P. 

Sectional 
Area  of  Flue 
per  square  ft. 
of  heating 
surface. 

Ibs. 

sq.  in. 

sq.  in. 

Bq.  in. 

sq.  ft. 

sq.  in. 

1 

10 

123 

106 

2 

20 

'iso 

191 

149 

15 

6-0 

8 

80 

189 

285 

188 

13 

4-8 

4 

40 

220 

270 

210 

11 

5-0 

6 

50 

808 

235 

6 

60 

'270 

881 

258 

10-7 

4-2 

7 

70 

858 

278 

8 

80 

'872 

8S8 

296 

10-2 

4-3 

9 

90 

406 

816 

10 

100 

"466 

428 

338 

10 

4-0 

11 

110 

468 

860 

12 

120 

'468 

469 

865 

9-8 

8-9 

18 

180 

488 

880 

14 

140 

'607 

607 

894 

9-8 

8-6 

15 

150 

524 

408 

16 

160 

'660 

541 

421 

9-7 

8-5 

17 

170 

554 

481 

18 

ISO 

'588 

675 

446 

9-8 

3-2 

19 

190 

590 

459 

20 

200 

'616 

606 

471 

10 

8-0 

80 

800 

720 

724 

577 

9-8 

2-4 

45 

450 

795 

909 

707 

9-6 

1-7 

60 

600 

1,049 

818 

75 

750 

1,178 

912 

100 

1,000 

1,866 

1,862 

1,059 

8 

16 

312  PEOPORTIONS    OF   STEAM-BOILERS. 

Mr.  Watt  reckoned  that  in  his  boilers  8  Ibs.  of  coal  would 
evaporate  a  cubic  foot  of  water  in  the  hour,  which  is  equivalent 
to  an  actual  horse-power  in  the  case  of  engines  working  without 
expansion.  Good  Welsh  coal,  however,  it  has  been  found,  will 
evaporate  10  Ibs.  of  water  for  each  pound  of  coal,  which  is 
equivalent  to  1*6  cubic  feet  of  water,  or  1*6  horse's  power  in  the 
case  of  an  engine  working  without  expansion ;  and  if  such  a 
measure  of  expansion  be  used  as  will  double  the  efficiency  of 
the  steam,  then  10  Ibs.  of  coal  burned  in  the  furnace  will  gene- 
rate 3 '2  actual  horses'  power.  To  attain  this  measure  of  effi- 
ciency, however,  the  steam  would  have  to  be  cut  off  between 
^  and  £  of  the  stroke,  and  in  the  best  boilers  and  engines  work- 
ing with  the  usual  rates  of  expansion  it  will  not  be  safe  to 
reckon  more  than  2  (or  at  most  2£)  actual  horses'  power  as  ob- 
tainable by  the  evaporation  of  a  cubic  foot  of  water.  When, 
therefore,  engines  work  up  to  five  times  their  nominal  power,  as 
they  now  often  do,  it  can  only  be  done  by  passing  through  them 
twice  the  quantity  of  steam  that  answers  to  their  nominal  power 
— or,  in  other  words,  by  making  the  boilers  of  twice  the  propor- 
tionate size,  unless  where  some  expedient  for  producing  an  ar- 
tificial draught  is  employed. 

The  proper  height  of  chimney  where  the  sectional  area 
of  the  flue  is  known  can  easily  be  deduced  from  the  foregoing 
rule. 

4/P  x  300  ,  (VP  x  300) 

For  if  A  =  -    — TT—  then  Ji  = r— 

yh  A 

which  formula  put  into  words  is  as  follows : — 

TO  FIND  THE  PEOPEE  HEIGHT  OF  A  CHIMNEY  IN  FEET  WHEN  THE 
NUMBEB  OF  POUNDS  OF  COAL  CONSUMED  PEE  HOUB  AND  ALSO 
THE  SECTIONAL  AEEA  OF  THE  FLUE  AEE  KNOWN. 

KTJLE. — Multiply  the  square  root  of  the  number  of  pounds  of 
coal  consumed  per  hour  by  the  constant  number  300,  and  di- 
vide the  product  by  the  sectional  area  of  the  flue  in  square 
inches  ;  the  square  of  the  quotient  is  the  proper  height  of  the 
chimney  in  feet. 


DIMENSIONS   OF   CHIMNEYS   FOR   GIVEN   POWERS.     313 

Example  1. — "What  is  the  proper  height  of  the  chimney  of  a 
hoiler  consuming  100  Ibs.  of  coal  per  hour,  and  with  a  sectional 
area  of  flue  of  428  square  inches. 

Here  4/100  =  10,  and  10  x  300  =  3000,  which  divided  by 
428  =  7,  the  square  of  which  is  49,  which  is  the  proper  height 
of  the  chimney  in  feet. 

Example  2. — "What  is  the  proper  height  of  the  chimney  of  a 
flue-boiler  consuming  100  Ibs.  of  coal  per  hour,  and  with  a  sec- 
tional area  of  flue  of  333  square  inches  ? 

Here  4/100  =  10,  and  10  x  300  =  3000,  which  divided  by 
333  =  9,  the  square  of  which  is  81,  which  is  the  proper  height 
of  the  chimney  in  feet. 

In  flue-boilers,  the  sectional  area  of  the  chimney  will  be  the 
same  as  that  of  the  flue  of  a  boiler  of  half  the  power.  Hence  in 
the  foregoing  Table  the  proper  sectional  area  of  the  chimney  of 
a  20-horse  boiler — the  chimney  being  49  feet  high— will  be  the 
same  as  the  sectional  area  of  the  flue  of  a  10-horse  boiler,  name- 
ly 428  square  inches,  with  a  height  of  chimney  of  49  feet ;  and 
the  proper  sectional  area  of  the  chimney  of  a  30-horse  boiler 
will  be  the  same  as  that  of  the  flue  of  a  15-horse  boiler,  namely, 
524  square  inches,  with  a  height  of  chimney  of  49  feet.  If  the 
chimney  be  91  feet  high,  then  the  values  will  become  333  and 
408  square  inches  respectively.  As  then  the  area  of  the  chimney 
should  be  the  same  as  that  of  the  flue  of  the  boiler  of  half  the 
power,  it  is  needless  to  give  a  separate  rule  for  finding  the  area 
of  the  chimney,  as  such  rule  will  be  in  all  respects  the  same  as 
that  for  finding  the  proper  area  of  the  flue,  except  that  we  take 
half  the  number  of  pounds  of  coal  burned  per  hour  instead  of 
the  whole. 

In  marine  tubular  boilers  the  total  capacity  or  bulk  of  the 
boiler,  exclusive  of  the  chimney,  is  about  8  cubic  feet  for  each 
cubic  foot  of  water  evaporated  per  hour — divided  in  the  propor- 
tion of  6 -5  cubic  feet  devoted  to  the  water,  furnaces,  and  tubes, 
and  1'5  cubic  foot  occupied  as  a  receptacle  or  repository  for  the 
bteam.  The  common  diameter  of  tube  in  marine  boilers  is  about 
8  inches,  and  the  length  is  28  or  30  times  the  diameter.  In  lo- 
comotive loilers  the  usual  diameter  of  the  tubes  is  2  inches,  and 
14 


314 


PROPORTIONS    OF    STEAM-BOILERS. 


the  length  is  about  60  times  the  diameter.  The  area  of  the  blast 
orifice  in  locomotives  is  about  TVth  of  the  area  of  the  chimney. 
The  fire-bars  are  commonly  £  inch  thick,  and  the  air-spaces  are 
made  1  inch  wide  for  fast  trains.  The  main  dimensions  of  ma- 
rine and  locomotive  boilers  required  for  the  evaporation  of  a 
cubic  foot  of  water,  are  given  in  the  following  Table : — 

PROPORTIONS     OF    MODERN    BOILERS    REQUIRED    TO    EVAPORATE  A 
CUBIC  FOOT   OF  WATER  PER  HOUR. 


Proportion  required  per  Cubic  Foot  evaporated 
per  hour. 

Marine  Flue. 

Marine  Tubu- 
lar. 

locomotive. 

Square  feet  of  heating  surface  

8 

9  to  10 

6 

70 

70 

18 

Square  inches  sectional  area  of  flue  or  tubes 
Square  inches  sectional  area  of  chminey.  .  . 
Square  feet  of  heating  surface  per  square 
foot  of  fire  grate  

13 
6 

16'48 

10 

7 
18-54 

81 

2-4 

48 

Pounds  of  coal  or  coke  consumed  on  each 
square  foot  of  fire  grate  per  hour.  

16 

16 

62 

The  quantity  of  coal  or  coke  burned  on  each  square  foot  of 
fire-grate  in  the  hour  to  evaporate  a  cubic  foot  of  water  will  of 
course  very  much  depend  on  the  goodness  of  the  coal  or  coke. 
In  the  above  Table  the  average  working  result  of '8  Ibs.  of  water 
evaporated  by  1  Ib.  of  coal,  or  a  cubic  foot  of  water  evaporated 
by  7 '8  Ibs.  of  coal,  is  taken. 

The  efficiency  of  a  steam  vessel  is  measured  by  the  expendi- 
ture of  fuel  necessary  to  transport  a  given  weight  at  a  given 
speed  through  a  given  space,  and  one  of  the  most  efficient  steam 
vessels  of  recent  construction  is  the  steamer  Hansa,  built  by 
Messrs.  Oaird  &  Co.,  to  ply  between  Bremen  and  America.  In 
this  vessel  there  are  two  inverted  direct-acting  engines,  with  cy- 
linders 80  inches  diameter  and  3|  feet  stroke.  There  are  four 
tubular  boilers,  with  four  furnaces  in  each,  containing  a  total 
grate  surface  of  350  square  feet,  and  a  heating  surface  of  9,200 
square  feet ;  besides  which  there  is  a  superheater,  containing  a 
heating  surface  of  2,100  square  feet.  The  steam  is  of  25  Ibs. 
pressure  on  the  square  inch,  and  it  is  condensed  by  being  dis- 
charged into  a  vessel  traversed  by  3,584  brass  tubes,  1  inch  ex- 
ternal diameter,  and  7  feet  long.  Each  tube  having  1'75  square 


SURFACE  FOR  GENERATING  AND  CONDENSING.   315 

feet  of  cooling  surface,  the  total  cooling  surface  will  be  6,272,  or 
about  two-thirds  of  the  amount  of  heating  surface.  The  cooling 
water  is  sent  through  the  tubes  by  means  of  two  double  acting 
pumps,  21  inches  diameter  and  2-t  inches  stroke,  worked  from 
the  forward  end  of  the  crank-shaft.  It  is  much  better  to  send 
the  water  through  the  tubes  than  to  send  the  steam  through 
them.  But  standing  and  hanging  bridges  of  plate-iron  should  be 
introduced  alternately  in  the  chamber  traversed  by  the  tubes,  so 
as  to  compel  the  current  of  steam  to  follow  a  zigzag  course ;  and 
the  steam  should  be  let  in  at  that  end  of  the  chamber  at  which 
the  water  is  taken  off,  so  that  the  hottest  steam  may  encounter 
the  hottest  water.  It  would  further  be  advantageous  to  inject 
the  feed  water  into  a  small  chamber  in  the  eduction-pipe,  so  as 
to  raise  the  feed-water  to  the  boiling-point  before  being  sent 
into  the  boiler ;  or  the  feed-pipe  might  be  coiled  in  the  eduction- 
pipe  so  as  to  receive  the  first  part  of  the  heat  of  the  escaping 
steam.  A  length  of  7  feet  appears  to  be  rather  great  for  a  pipe 
an  inch  diameter,  as  the  water  at  the  end  of  it  will  become  so 
hot  as  to  cease  to  condense  any  steam,  unless  the  velocity  of  the 
flow  be  so  great  as  to  involve  considerable  resistance  from  fric- 
tion. Short  pipes,  with  an  abundant  supply  of  cold  water,  will 
enable  a  very  moderate  amount  of  refrigerating  surface  to  suffice, 
as  plainly  appears  from  Mr.  Joule's  experiment,  already  recited. 
If  we  reckon  the  engines  of  the  Hansa  at  TOO  horses'  power, 
there  will  be  half  a  square  foot  of  grate-bars  per  nominal  horse- 
power, and  13*1  square  feet  of  heating  surface  per  nominal 
horse-power  in  the  boiler,  besides  3  square  feet  in  the  super- 
heater, making  in  all  16'1  square  feet  of  heating  surface  per 
nominal  horse-power,  or  32 '2  square  feet  of  heating  surface  per 
square  foot  of  fire-grate.  If  we  take  9  square  feet  as  evapora- 
ting a  cubic  foot  of  water  per  hour,  then  the  total  evaporation 
of  the  boilers  in  cubic  feet  will  be  9,200  -s-  9  =  1,022  cubic  feet 
per  hour ;  and  if  we  reckon  8  Ibs.  of  coal  as  necessary  to  evapo- 
rate a  cubic  foot,  then  the  consumption  of  coal  per  hour  will  bo 
8,176  Ibs,  or  3'6  tons  per  hour,  supposing  the  boiler  to  be  work- 
ing at  its  greatest  power.  This  is  11*6  Ibs.  of  coal  per  nominal 
horse-power,  reckoning  the  power  at  700 ;  and  at  this  rate  of 


316  PROPORTIONS    OF   STEAM-BOILERS. 

consumption  23'2  Ibs.  of  coal  will  be  burned  every  hour  on  each 
square  foot  of  fire-grate,  to  generate  the  steam  rqeuired  for  a 
nominal  horse-power,  or  it  will  be  16  Ibs.  on  each  square  foot 
every  hour  to  evaporate  a  cubic  foot — there  being  nearly  T5 
cubic  feet  of  water  evaporated  for  the  production  of  each  nomi- 
nal horse-power. 

INDICATIONS   TO   EE   FULFILLED   IN   MAKING  BOILERS. 

In  all  boilers  the  expedients  for  maintaining  a  proper  circu- 
lation of  the  water,  so  that  the  flame  may  act  upon  solid  water, 
and  not  upon  a  mixture  of  water  and  steam,  have  been  greatly 
neglected ;  and  the  consequence  is  that  a  much  larger  amount  of 
surface  is  required  than  would  otherwise  be  necessary.  The 
metal  of  the  boiler  is  often  bent  and  buckled  by  being  overheated, 
and  priming  takes  place  to  an  inconvenient  extent.  In  all  tubu- 
lar boilers  the  water  should  be  within  the  tubes,  and  those  tubes 
should  be  vertical,  so  as  to  enable  the  current  of  steam  and  water 
to  rise  upward  as  rapidly  as  possible.  The  best  form  of  steam- 
boat boiler  hitherto  introduced  is  the  haystack  boiler,  for  which 
we  are  indebted  to  the  fertile  ingenuity  of  Mr.  David  Napier, 
and  in  which  boiler  the  prescribed  indications  are  well  fulfilled. 
In  the  haystack  boiler,  which  is  much  used  in  the  smaller  class 
of  river-boats  on  the  Clyde — but  which,  like  the  oscillating  en- 
gine at  the  earlier  period  of  its  history,  has  not  yet  been  employed 
in  seagoing  vessels — the  tubes  are  vertical,  with  the  water  within 
them ;  and  the  smoke  on  its  way  to  the  chimney  imparts  its  heat 
to  the  water  by  impinging  upon  the  outsides  of  the  tubes.  The 
late  Lord  Dundonald  (another  remarkable  mechanical  genius) 
proposed  a  similar  plan  of  boiler ;  and  boilers  on  his  principle 
— in  which  the  furnace  flue  of  a  common  marine  flue-boiler  is 
filled  with  a  grove  of  small  vertical  tubes  on  which  the  smoke 
impinges  on  its  way  to  the  chimney — have  been  much  used  on 
the  Continent  with  good  results,  and  were  also  introduced  in  the 
Collins  line  of  steamers  navigating  the  Atlantic.  The  Clyde 
haystack  boilers  are  generally  made  of  the  form  of  an  upright 
cylinder  with  a  hemispherical  top,  from  the  centre  of  which  the 
chimney  ascends.  The  furnace  is  circular,  with  a  water-space 


NAPIER'S  AND  DTJNDONALD'S  BOILERS.  317 

all  around  it,  and  with  a  circular  crown ;  so  that  the  furnace 
forms,  in  fact,  a  short  cylinder,  divided  in  some  cases  into  four 
quarters  by  vertical  water-spaces  crossing  one  another.  Suitable 
passages  are  provided  to  conduct  the  smoke  from  the  furnace 
into  a  cylindrical  chamber  situated  above  it — the  diameter  of 
this  cylinder  being  the  same  as  that  of  the  shell  of  the  boiler, 
less  the  breadth  of  a  water-space  which  runs  round  it ;  and  the 
height  of  this  cylinder  being  equal  to  the  length  of  the  tubes. 
The  tubes  are  set  in  circles  round  the  chimney ;  and  the  smoke, 
which  is  delivered  from  the  furnace  near  the  exterior  of  the 
cylindrical  chamber,  has  to  make  its  way  among  the  vertical 
tubes  before  it  can  reach  the  chimney.  The  lower  tube-plate 
and  the  furnace  crown  are  stayed  to  one  another  by  frequent 
bolts,  and  the  cylindrical  chamber  containing  the  tubes  is  also 
bolted  at  intervals  to  the  shell  of  the  boiler.  The  water-space 
intervening  between  the  lower  tube-plate  and  furnace  crown  is 
made  very  wide,  so  as  to  hold  a  large  body  of  water,  and  also  to 
enable  a  person  to  reach  in  should  repairs  be  required.  The  only 
weak  part  of  this  boiler  is  the  root  of  the  chimney,  which  some- 
times has  collapsed  from  becoming  overheated  by  the  flame  as- 
cending the  chimney  before  the  steam  has  been  generated ;  and 
the  small  pressure  of  the  air  shut  within  the  boiler  when  heated 
has  caused  the  root  of  the  chimney  to  collapse.  This  risk  is 
easily  prevented  by  placing  several  rings  of  T-iron  around  the 
root  of  the  chimney,  within  the  steam-chest,  and  also  by  carry- 
ing down  the  plating  of  the  chimney  for  some  distance  into  the 
tube-chamber,  so  as  to  constitute  a  hanging-bridge  that  would 
hinder  the  hottest  part  of  the  smoke  from  escaping,  and  retain  it 
in  the  tube-chamber,  until  it  had  given  out  the  principal  part  of 
its  heat  to  the  water.  In  all  boilers  of  this  construction  these 
precautions  should  be  adopted ;  and  it  would  further  be  useful 
to  place  a  short  piece  of  pipe  in  the  mouth  of  every  upright  tube, 
so  as  to  continue  the  tube  up  to  the  water-level,  whereby  the 
column  being  elongated  its  ascensional  force  would  be  increased, 
and  the  circulation  of  the  water  be  rendered  more  rapid. 

As  this  species  of  boiler  is  likely  to  come  into  use  both  for 
steam-vessels  and  for  locomotives,  it  will  be  proper  to  indicate 


318  PROPORTIONS   OF   STEAM-BOILERS. 

the  forms  which  appear  to  be  most  suitable  for  those  objects.  In 
steam-vessels  it  is  desirable  to  combine  the  introduction  of  a  spe- 
cies of  boiler  adapted  for  working  at  a  higher  pressure,  with 
arrangements  for  burning  the  smoke,  which  will  be  best  done 
by  maintaining  a  high  temperature  in  the  furnace ;  and  a  high 
degree  of  heat  will  be  best  kept  up  in  the  furnace  by  forming  it 
of  firebrick  instead  of  surrounding  it  with  water  in  the  usual 
manner.  If,  therefore,  a  square  box  of  iron  be  taken  and  lined 
with  firebrick,  and  if  it  be  divided  longitudinally  and  transversely 
by  these  brick  walls,  and  afterwards  be  arched  over,  we  shall 
have  four  furnaces,  requiring  merely  the  introduction  of  the  fire- 
bars to  enable  them  to  be  put  into  operation.  Suppose  that  on  the 
top  of  each  of  these  square  boxes  a  barrel  of  vertical  tubes  is 
placed,  the  barrel  being  sufficiently  sunk  into  the  brickwork  to 
establish  a  communication  for  the  smoke  between  a  hole  at  each 
of  the  four  top  corners  of  the  box  and  corresponding  perforations 
in  the  barrel,  we  shall  then  have  the  smoke  from  each  of  the 
four  furnaces  into  which  the  box  is  divided  escaping  from  one 
corner  into  the  chamber  containing  the  tubes,  and  after  travelling 
among  them  passing  to  the  chimney.  In  such  a  boiler  the  circu- 
lation of  the  water  could  be  maintained  by  forming  the  external 
water-space  very  thick,  and  by  placing  a  diaphragm-plate  in  it ;  so 
that  the  water  and  steam  could  rise  upward  on  the  side  of  the 
water-space  next  to  the  tube  chamber,  while  the  solid  water  de- 
scended on  that  side  of  the  water-space  next  to  the  boiler-shell. 
The  intervening  plate  would  enable  these  currents  to  flow  in 
opposite  directions  without  interfering  with  one  another. 

In  a  boiler  of  this  kin  d  the  grate-bars  should  have  a  sufficient 
declivity  to  enable  the  coal  to  advance  itself  spontaneously  upon 
them ;  and  if  there  are  two  lengths  of  firebars  in  the  furnace,  the 
front  length  should  be  set  closer  together  than  the  others,  so  as 
partially  to  coke  the  coal  as  on  a  dead-plate,  before  it  enters  into 
combustion.  This  coking  would  be  affected  by  the  radiant  heat 
of  the  furnace,  to  which  heat  the  coal  would  be  exposed.  The 
openings  through  which  the  smoke  would  escape  to  the  tube- 
chamber  might  be  perforations  or  lattice  openings  in  the  brick- 
work, BO  as  to  bring  every  particle  of  the  smoke  into  intimate 


IMPORTANCE   OF  RAPID   CIRCULATION.  319 

contact  with  the  incandescent  material  of  which  the  furnace  is 
composed;  and  these  perforations  should  not  have  too  much 
area,  else  the  heat  would  escape  to  the  tubes  too  rapidly,  and 
the  temperature  of  the  furnace  would  fall.  To  maintain  a  suffi- 
cient draught  to  bring  in  the  requisite  supply  of  air  to  the  fuel, 
a  jet-pipe  of  steam  could  be  introduced  at  the  bottom  of  the 
chimney;  which  jet-pipe  would  open  into  a  short  piece  of  pipe 
of  larger  diameter,  also  pointing  up  the  chimney,  and  it  into 
another  larger  piece,  and  so  on.  The  jet  at  each  of  these  short 
pieces  of  pipe  would  draw  in  smoke  and  form  with  the  previous 
jet  a  new  jet,  which  would  become  of  larger  and  larger  volume 
and  less  velocity  at  successive  steps,  until  the  dimensions  of  the  jet 
had  enlarged  to  an  area  perhaps  equal  to  half  the  area  of  the 
chimney.  It  will  be  sufficient  if  the  length  of  each  piece  of  pipe 
be  a  little  greater  than,  its  diameter ;  and  the  lower  end  of  each 
piece,  or  that  end  facing  the  current  of  smoke,  should  be  opened 
a  little  into  a  funnel  shape,  the  better  to  catch  the  smoke  and 
carry  it  forward,  to  form  with  the  steam  a  jet  continually  en- 
larging its  dimensions.  By  this  mode  of  construction  a  powerful 
draught  will  be  created  by  the  jet  with  a  very  small  expenditure 
of  steam.  The  area  through  the  cylindrical  hanging-bridge  at 
the  root  of  the  chimney  should  not  be  large,  and  the  bridge  itself 
should  be  perforated  with  holes  in  some  places,  so  as  to  establish 
a  sufficient  current  of  the  smoke  upward  among  the  tubes  to  pre- 
vent the  heat  and  flame  being  swept  past  direct  to  the  bottom  of 
the  chimney  without  rising  among  the  tubes  to  impart  its  heat 
to  them. 

In  the  case  of  locomotive  boilers  formed  with  upright  tubes, 
the  fire-box  would  be  the  same  as  at  present;  but  that  part  of 
the  boiler  called  the  barrel,  and  which  is  now  filled  with  longi- 
tudinal tubes,  would  be  formed  with  flat  sides  and  bottom  and  a 
semicircular  top,  so  that  it  would  have  the  same  external  form 
as  the  external  fire-box,  and  this  vessel  would  be  traversed  by  a 
square  flue,  in  which  the  vertical  tubes  would  be  set.  The  sides 
and  bottom  of  this  flue  would  be  affixed  to  the  shell  by  staybolts 
in  the  same  manner  as  the  internal  and  external  fire-boxes  are 
stayed  to  one  another ;  and  the  top,  being  semicircular,  would 


320 


PROPORTIONS    OF    STEAM-BOILERS. 


not  require  staying,  while  the  upper  tube-plate  forming  the  top 
of  the  square  internal  flue  would  be  strutted  asunder  and  prevent- 
ed from  collapsing  by  the  tubes  themselves,  some  of  which  should 
be  screwed  into  the  plates  or  formed  with  internal  nuts,  to  make  • 
them  more  efficient  in  this  respect.  Such  a  boiler  would  have 
various  advantages  over  ordinary  locomotive  boilers,  and  might 
be  made  of  any  power  that  was  desired  without  any  limitation 
being  imposed  by  the  width  of  the  gauge  of  the  railway.  Such 
boilers  might  also  be  used  for  steam-vessels  by  merely  increasing 
the  area  of  the  fire-grate. 

8TBENGTH  OF  BOILEE3. 

The  proportions  which  a  boiler  should  possess  in  order  to  have 
a  safe  amount  of  strength  will  be  determined  partly  by  the  pres- 
sure of  the  steam  within  the  boiler,  and  partly  by  the  dimen- 
sions and  configuration  of  the  boiler  itself.  The  best  propor- 
tions of  the  riveted  joints  of  the  plates  of  which  boilers  are  made 
are  as  follows : — 

BEST   PBOPOBTION8   OF   EIVETED   STEAM-TIGHT   JOINTS. 


Thickness  of 
Plate  In  Inches. 

Proper 
Diameter  of 
Rivets  in 

Proper  Length 
in  inches  of 
Rivets   from 

Proper  distance 
from  Centre 
to  Centre  of 

Proper 
Quantity  of 
Lap  in  inches 
In  Single 

Proper 
Quantity  of 
Lap  in  inches 
in  Double 

Inches. 

Head. 

Rivets    in 

Riveted 

Riveted 

inches. 

Joints. 

Joints. 

A 

1 

i 

H 

H 

2^ 

i 

i 

1* 

it 

l£ 

» 

A 

} 

H 

it 

H 

H 

t 

f 

If 

i* 

2rV 

H 

1 

it 

2* 

2 

ii 

»t 

i 

it 

2* 

a* 

2J 

4& 

3. 

H 

»i 

3 

»! 

gj 

If  the  strength  of  the  plate  iron  be  taken  at  100,  then  it  has 
been  found  experimentally  that  the  strength  of  a  single-riveted 
joint  will  be  represented  by  the  number  56,  and  a  double  riveted 
joint  by  the  number  70.  According  to  the  experiments  of  Messrs. 
Napier  and  Sons,  the  average  tensile  strength  of  rolled  bars  of 
Yorkshire  iron  was  found  to  61,505  Ibs.  per  square  inch  of  section, 


STRAINS  AND  STRENGTHS.  321 

and  the  average  strength  of  bars  made  by  nine  different  makers 
(and  purchased  promiscuously  in  the  market)  was  found  to  be 
69,276  Ibs.  per  square  inch  of  section.  The  tensile  strength  of 
1  cast  steel  bars  intended  for  rivets  was  found  to  be  106,950  Ibs. 
per  square  inch  of  section,  of  homogeneous  iron  90,647  Ibs.,  of 
forged  bars  of  puddled  steel  71,486  Ibs.  and  of  rolled  bars  of 
puddled  steel  70,166  Ibs.  per  square  inch  of  section.  The  strength 
of  Yorkshire  plates  Messrs.  Napier  found  to  be — lengthwise 
55,433  Ibs.,  crosswise  50,462  Ibs.,  and  the  mean  was  52,947  Ibs. 
per  square  inch  of  section.  The  tensile  strength  of  ordinary  best 
and  lest-lest  boiler  plates,  as  manufactured  by  ten  different 
makers,  was  found  to  be — lengthwise  50,242  Ibs.,  crosswise  45,986 
Ibs.,  and  the  mean  was  48,114  Ibs.  per  square  inch  of  section. 
Plates  of  puddled  steel  varied  from  85,000  Ibs.  to  101.000  Ibs. 
per  square  inch  of  section,  and  homogeneous  iron  was  found 
to  have  a  tensile  strength  of  about  96,000  Ibs.  per  square  inch  of 
section. 

Experiments  have  been  made  to  determine  the  strength  of 
bolts  employed  to  stay  the  flat  surfaces  of  boilers  together ;  and 
it  has  been  found  that  an  iron  bolt  f  ths  of  an  inch  diameter,  liko 
the  staybolt  of  a  locomotive,  screwed  into  a  copper  plate  f  ths 
of  an  inch  thick,  and  not  riveted,  bore  a  strain  of  18,260  Ibs. 
before  it  was  stripped  and  drawn  out.  When  the  end  of  the  bolt 
was  riveted  over  it  bore  24,140  Ibs.  before  giving  way,  when  the 
head  of  the  rivet  was  torn  off,  and  the  bolt  was  stripped  and 
drawn  through  the  plate.  When  the  bolt  was  screwed  into  an 
iron  plate  fths  of  an  inch  thick,  and  the  head  riveted  as  before, 
it  bore  a  load  of  28,760  Ibs.  before  giving  way,  when  the  stay  was 
torn  through  the  middle.  When  the  staybolt  was  of  copper 
screwed  into  copper  plate  and  riveted,  it  broke  with  a  load  of 
16,265  Ibs.,  after  having  first  been  elongated  by  the  strain  one- 
sixth  of  its  length.  Locomotive  fire-boxes  are  usually  stayed 
with  f-inch  bolts  of  iron  or  copper  pitched  4  inches  asunder,  and 
tapped  into  the  metal  of  the  outer  and  inner  fire-boxes,  and  the 
stays  are  generally  screwed  from  end  to  end.  These  stays  give  a 
considerable  excess  of  strength  over  the  shell,  but  it  is  necessary 
to  provide  for  the  risk  of  a  bad  bolt. 
14* 


322  PROPORTIONS   OF   STEAM-BOILERS. 

"With  these  data  it  is  easy  to  tell  what  the  scantlings  of  a 
boiler  should  be  to  withstand  any  given  pressure.  If  we  take  the 
strength  of  a  single-riveted  joint  at  34,000  Ibs.  per  square  inch, 
then  in  a  cylindrical  boiler  the  bursting  strength  in  pounds  will 
be  measured  by  the  diameter  of  the  boiler  in  inches  multiplied 
by  twice  the  thickness  of  the  plate  in  inches,  and  by  the  pressure 
of  the  steam  per  square  inch  La  pounds ;  and  this  product  will  be 
34,000  Ibs.  Thus  in  a  cylindrical  boiler  3  feet  or  36  inches  diame- 
ter and  half  an  inch  thick,  if  we  suppose  a  length  of  one  inch  to  be 
cut  off  the  cylinder  we  shall  have  a  hoop  |  an  inch  thick  and  1  inch 
long.  If  we  suppose  one-half  of  the  hoop  to  be  held  fast  while 
the  steam  endeavours  to  burst  off  the  other  half,  the  separation 
will  be  resisted  by  two  pieces  of  plate  iron  1  inch  long  and  -J  an 
inch  thick ;  or,  in  other  words,  the  resisting  area  of  metal  will  be 
one  square  inch,  to  tear  which  asunder  requires  34,000  Ibs.  The 
separating  force  being  the  diameter  of  the  boiler  in  inches  mul- 
tiplied by  the  pressure  of  the  steam  on  each  square  inch,  and  this 
being  equal  to  34,000  Ibs.,  it  follows  that  if  we  divide  the  total 
separating  force  in  pounds  by  the  diameter  in  inches,  we  shall 
obtain  the  pressure  of  the  steam  on  each  square  inch  that  would 
just  burst  the  boiler.  N"ow  34,000  divided  by  36  (which  is  the 
diameter  of  the  boiler  in  inches)  gives  944'4  Ibs.  as  the  pressure 
of  the  steam  on  each  square  inch  that  would  burst  the  boiler.  A 
certain  proportion  of  the  bursting  pressure  will  be  the  safe  work- 
ing pressure,  and  Mr.  Fairbairn  considers  that  one  sixth  of  the 
bursting  pressure  will  be  a  safe  working  pressure ;  but  in  my 
opinion  the  working  pressure  should  not  be  greater  than  between 
one-seventh  and  one-eighth  of  the  bursting  pressure.  The 
rule  which  I  gave  in  my  '  Catechism  of  the  Steam  Engine,' 
for  determining  the  proper  thickness  of  a  single-riveted  boiler, 
proceeds  on  the  supposition  that  the  working  pressure  should  be 
TJg-  of  the  bursting  pressure.  That  rule  is  as  follows : — 

TO   FEND   THE   PEOPEE  THICKNESS   OF   THE   PLATES   OF   A   SINOLE- 
EIVETED   CYLIKDBICAL  BOILEE. 

RULE. — Multiply  the  internal  diameter  of  the  toiler  in  inches 
~by  the  pressure  of  the  steam  in  Ibs.  per  square  inch  above  the 


STRAINS   AND    STRENGTHS.  323 

atmosphere,  and,  divide  the  product  ~by  8,900:  the  quotient  is 
the  proper  thickness  of  the  plate  of  the  boiler  in  inches. 

Example  1. — "What  is  the  proper  thickness  of  the  plating 
of  a  single-riveted  cylindrical  boiler  of  3J  feet  diameter,  and 
intended  to  work  with  a  pressure  of  80  Ibs.  on  the  square 
inch? 

Here  42  inches  (which  is  the  diameter)  multiplied  by 
80  =  3360,  and  this  divided  by  8900  =  '377,  or  a  little  over  f  of 
an  inch.  The  decimal  '375  is  f  of  an  inch. 

Example  2. — What  is  the  proper  thickness  of  a  single-riveted 
cylindrical  boiler  3  feet  diameter,  intended  to  carry  a  pressure 
of  100  Ibs.  on  the  square  inch  ? 

Here  36  inches  x  100  =  3600,  which  divided  by  8900  =  '4, 
or,  as  nearly  as  possible,  ^  and  •£%. 

As  the  double-riveted  joint  is  stronger  than  the  single-riveted 
in  the  proportion  of  70  to  56,  it  follows  that  56  square  inches  of 
sectional  area  in  a  double-riveted  boiler  will  be  as  strong  as  70 
square  inches  in  a  single-riveted.  This  relation  is  expressed  by 
the  following  rule : — 

TO   FIND   THE   PEOPEB   THICKNESS   OF   THE   PLATES   OF   A   DOTJBLE- 
EIVETED   CYLINDRICAL   BOILEB. 

RULE. — Multiply  the  internal  diameter  of  the  boiler  in  inches  by 
the  pressure  of  the  steam  in  pounds  per  square  inch  above  the 
atmosphere,  and,  divide  the  product  by  the  constant  number 
11140 :  the  quotient  will  be  the  proper  thickness  of  the  boiler 
in  inches  when  the  seams  are  double-riveted. 

Example  1. — What  is  the  proper  thickness  of  the  plates 
of  a  double-riveted  cylindrical  boiler  42  inches  diameter,  and 
intended  to  work  with  a  pressure  of  80  Ibs.  per  square  inch  ? 

Here  42  x  80  =  3360,  and  this  divided  by  11140  = '3016, 
or  about  ^  of  an  inch,  which  is  the  proper  thickness  of  the 
plates  when  the  boiler  is  double-riveted. 

Example  2. — What  is  the  proper  thickness  of  a  double-riveted 
cylindrical  boiler  8  feet  diameter,  intended  to  carry  a  pressure  of 
100  Ibs.  on  the  square  inch? 


324:  PROPORTIONS   OF   STEAM-BOILERS. 

Here  36  inches  x  100  =  3600,  which  divided  by  11140  =  -322, 
or  a  little  more  than  -fs  of  an  inch,  which  will  be  the  proper 
thickness  of  the  plates  of  the  boiler  when  the  seams  are  double- 
riveted. 

If  T  =  the  thickness  of  the  plate  in  inches,  D  =  the  diameter 
of  the  cylinder  or  shell  of  the  boiler  in  inches,,  and  P  =  the 
pressure  of  the  steam  per  square  inch  :  Then 

D  P 

is  the  formula  for  the  thickness  of  single-riveted 


is  the  formula  for  double-riveted  boilers. 


boilers,  and 

DP 


11140 
Moreover,  in  single-riveted  boilers — 


p  — 


p 

8900  T 


D 

So  also  for  double-riveted  boilers — 


p 
p  _  11140  T 

D 

These  formulas  put  into  words  are  as  follows : — 

TO  FIND  THE  PEOPEB  DIAMETEE  OF  A  eiNGLE-EIVETED  BOILEE 
OF  KNOWN  THICKNESS  OF  PLATES  AND  KNOWN  PRESSURE  OF 
STEAM. 

RULE. — Multiply  the  thickness  in  inches  T>y  the  constant  number 
8900,  and  divide  ly  the  pressure  of  the  steam  in  tts.  per  square 
inch.     The  quotient  is  the  proper  diameter  of  the  boiler  in 
inches. 
JZcample  1. — What  is  the  proper  diameter  of  a  single-riveted 

cylindrical  boiler  composed  of  plates  -377  inches  thick,  and 

intended  to  work  with  a  pressure  of  80  Ibs.  on  the  square 

inch? 

Here   -377  x  8900  =  3355-3,  which  divided  by  80=41-94 

inches,  or  42  inches  nearly,  which  is  the  proper  diameter  in  inches. 


STRAINS  AND  STRENGTHS.  325 

Example  2. — What  is  the  proper  diameter  of  a  single-riveted 
boiler  composed  of  plates  '4  inches  thick,  and  intended  to  work 
with  a  pressure  of  100  Ibs.  on  the  square  inch  ? 

Here  '4  x  8900  =  3560,  which  divided  by  100  =  35'6  inches, 
which  is  the  proper  diameter  of  the  cylindrical  shell  of  the  boiler 
in  this  case. 

TO  FIND  THE  PEESSTJEE  TO  WHICH  A  SINGLE-EFVETED  CYLINDBIOAL 
BOILEE  MAY  BE  WOEKED  WHEN  ITS  DIAMETEB  AND  THE  THICK- 
NESS OF  ITS  PLATING  AEE  KNOWN. 

RULE. — Multiply  the  thickness  of  the  plating  in  inches  7>y  the 

constant  number  8900,  and  divide  the  product  ty  the  diameter 

of  the  toiler  in  inches.     The  quotient  is  the  pressure  of  steam 
per  square  inch  at  which  the  boiler  may  Tie  worked. 

Example  1. — What  is  the  highest  safe-working  pressure  in  a 
single-riveted  boiler  42  inches  diameter,  and  composed  of  plates 
•377  of  an  inch  thick? 

Here  -377  x  8900  =  3355'3,  which  divided  by  42  =  79-8 
Ibs.  per  square  inch,  which  is  the  highest  safe  pressure  of  the 
steam. 

Example  2. — What  is  the  highest  safe- working  pressure  in  the 
case  of  a  single-riveted  boiler  36  inches  diameter,  and  composed 
of  plates  -4  of  an  inch  thick  ? 

Here  -4  x  8900  =  3560,  which  divided  by  36  =  99  Ibs.  per 
square  inch. 

The  rules  for  double-riveted  boilers  are  in  every  case  the 
same  as  those  for  single-riveted,  only  that  the  constant  11140  is 
used  instead  of  the  constant  8900.  It  will  therefore  be  unneces- 
sary to  repeat  the  examples  for  the  case  of  double-riveted  boilers. 

Mr.  Fairbairn  has  given  the  following  table  as  exhibiting  the 
bursting  and  safe-working  loads  of  single  riveted  cylindrical 
boilers.  But  I  have  already  stated  that  I  consider  Mr.  Fairbairn's 
margin  of  safety  too  small.  The  working  pressure,  however, 
which  he  gives  for  single-riveted  boilers  would  not  be  too  great 
for  double-riveted  boilers,  as  will  appear  by  comparing  those 
pressures  with  the  pressures  which  the  foregoing  rules  indicate 
may  bo  safely  employed. 


326 


PROPORTIONS    OF    STEAM-BOILERS. 


TABLE   SHOWING  THE   BURSTING   AND   SAFE-WORKING   PRESSURE 
OF    CYLINDRICAL   BOILERS,    ACCORDING   TO   MR.    FAIRBAIRN. 


Diameter  of 
Boiler. 

Working  pressure 
for  J  -inch  plates. 

Bursting  pressure 
for  .»  -inch  plates. 

Working  pressure 
for  %-inch  plates. 

Bursting  pressure 
for  ,^-lnch  plates. 

ft.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

3    0 

118 

708} 

157} 

944} 

3    8 

109 

653} 

145} 

871} 

8    6 

101 

607 

134} 

809} 

3    9 

94 

566} 

125} 

755} 

4    0 

86} 

531 

118 

708} 

4    3 

as* 

500 

111 

666} 

4    6 

78} 

472 

104} 

629} 

4    9 

74} 

447} 

99} 

596} 

5    0 

70} 

425 

94} 

566} 

5    3 

67} 

404} 

89} 

539} 

5    6 

64} 

886} 

85} 

515 

5    9 

61} 

363} 

82 

492} 

6    0 

59 

354 

78} 

472 

6    3 

56} 

340 

75} 

453} 

6    6 

54} 

326} 

72} 

435} 

6    9 

52} 

814} 

69} 

419} 

7    0 

50} 

308} 

67} 

404} 

7    8 

48} 

293 

65 

396} 

7    6 

47 

283} 

62} 

377} 

7    9 

45} 

274 

60} 

365} 

8    0 

44 

265} 

59 

354 

8    8 

42} 

257} 

57 

343} 

8    6 

41} 

260 

55} 

833} 

8    9 

40} 

242} 

54 

323} 

9    0 

89} 

236 

52} 

314} 

9    6 

37 

223} 

49} 

298} 

10    0 

85} 

212} 

47 

283} 

It  will  be  useful  to  compare  some  of  the  figures  of  this  tablo 
with  the  results  given  by  the  rules  just  recited.  For  example, 
according  to  Mr  Fairbairn,  a  single-riveted  boiler,  5  feet  diame- 
ter, and  formed  of  £-inch  plates,  may  be  habitually  worked  with 
safety  to  a  pressure  of  Q4^  Ibs.  on  the  square  inch.  Now,  by  our 
rule,  -5  x  8900  =  4450,  which  divided  by  60,  the  diameter  of  the 
boiler  in  inches,  gives  74  Ibs.  as  the  safe  pressure  at  which  the 
boiler  may  be  worked.  If  the  boiler  be  double-riveted,  then  we 
have  '5  x  11140  =  5570,  which,  divided  by  60,  gives  93  Ibs.  as 
the  pressure  per  square  inch  at  which  the  boiler  may  be  safely 
worked.  This  differs  very  little  from  Mr.  Fairbairn's  result  of 
94J  Ibs.,  and  his  table  may  therefore  be  used  if  the  results  be  re- 
garded as  applicable  to  double-riveted  boilers,  but  as  applied  to 
single-riveted  boilers  his  proportions,  I  consider,  are  too  weak. 
The  following  diameters  of  boilers  with  the  corresponding  thick- 


COLLAPSING   PRESSURE    OF   FLUES.  327 

ness  of  plates,  it  will  be  seen,  are  all  of  equal  strengths,  their 
bursting  pressure  being  450  Ibs.  per  square  inch,  which  answers 
to  34,000  Ibs.  per  square  inch  of  section  of  the  iron.  Diameter 
3  ft.,  thickness  -250  inches  ;  3£  ft.,  -291  ;  4  ft,,  '333  ;  4i  ft.,  -376  ; 
5  ft.,  -416;  5$  ft.,  458  ;  6  ft.,  -500;  6|  ft.,  -541  ;  7  ft.,  -583  ;  7i 
ft.,  -625  ;  and  8  ft.,  -666. 

The  collapsing  pressure  of  cylindrical  flues  follows  a  different 
law  from  the  bursting  pressure,  being  dependent,  not  merely 
upon  the  diameter  and  thickness  of  the  tube,  but  also  upon  its 
length  ;  and  Mr.  Fairbairn  gives  the  following  formula  for  com- 
puting the  collapsing  pressure.  If  T  =  the  thickness  of  tlie  iron, 
p  =  collapsing  pressure  in  Ibs.  per  square  inch,  L  =  length  of 
tube  in  feet,  and  D  =  diameter  of  tube  in  inches  ;  then 

T2-19 

p  =  806,300  f  - 

LD 

and  as  to  multiply  the  logarithm  of  any  number  is  equivalent  to 
raising  the  natural  number  to  the  power  which  the  logarithm  rep- 
resents, we  may  for  T2'19  write  2*19  log.  T.  With  this  trans- 
formation the  equation  becomes 


=  806,300 


If  now  we  take  the  thickness  of  the  plate  of  the  circular  flue  at 
•291  inches,  and  if  we  make  the  diameter  of  the  flue  12  inches 
and  its  length  10  feet,  the  equation  will  become 

P  =  806,800  gl19**'291. 

Now  '291  being  a  number  less  than  unity,  the  index  of  its  loga- 
rithm will  be  negative,  and  for  such  a  number  as  '291  the  index 
will  be  1,  the  minus  being  for  the  sake  of  convenience  written 
on  the  top  of  the  figure;  whereas  for  such  a  number  as  -0291  the 
index  will  be  2  ;  for  -00291  the  index  will  be  3,  and  so  on.  It 
does  not  signify,  so  far  as  the  index  is  concerned,  what  the  sig- 
nificant figures  are,  but  only  at  what  decimal  place  they  begin  ; 
and  -1  has  the  same  index  as  -291,  and  -01  as  -0291.  Now  the 
logarithm  of  291,  as  found  in  the  logarithmicjtables,  is  463893,  and 
the  index  being  "T,  the  whole  logarithm  is  1-463893.  In  multi- 


328 


PROPORTIONS    OF    STEAM-BOILERS. 


plying  a  logarithm  with  a  negative  index,  as  it  is  the  index  alone 
that  is  negative,  while  the  rest  of  the  logarithm  is  positive,  we 
must  multiply  the  quantities  separately,  and  then  adding  the 
positive  and  negative  quantities  together,  as  we  would  add  a 
debt  and  a  possession,  we  give  the  appropriate  sign  to  that 
quantity  which  preponderates.  Now  "463893  multiplied  by 
2-19  =  1-01592567,  and  1  multiplied  by  2-19  gives  2-19,  which  is 
a  negative  quantity.  Adding  these  products  together,  we  in 
point  of  fact  subtract  the  2'19  from  the  1'01592567,  which  leaves 
2'82592567.  Now  if  we  turn  to  the  logarithmic  tables,  we  shall 
find  that  the  number  answering  to  the  logarithm  82592567,  or 
the  number  answering  to  the  nearest  logarithm  thereto  (which  is 
825945),  is  6698;  but  as  the  index  is  negative,  this  quantity 
will  be  a  fraction,  and  the  index  being  2,  the  number  will  begin 
in  the  second  place  of  decimals — or,  in  other  words,  it  will  be 
0-C698.  Now  806300  multiplied  by  '06698  =  54004-974,  which, 
divided  by  120,  gives  450  Ibs.  as  the  collapsing  pressure.  If  we 
allow  the  same  excess  of  strength  to  resist  collapse  that  we 
allowed  to  resist  bursting — namely,  7'6  times— a  tube  of  the 
dimensions  we  have  supposed  will  be  safe  in  working  at  a  pres- 
sure of  60  Ibs.  on  the  square  inch.  But  the  strength  of  tubes  to 
resist  collapse  may  easily  be  increased  by  encircling  them  with 
rings  of  T  iron  riveted  to  the  tube.  Cylindrical  flues  of  different 
dimensions,  but  of  equal  strength  to  resist  collapse,  are  specified 
in  the  following  table : — 

CYLINDRICAL   FLUES   OF   EQUIVALENT   STRENGTH,    THE   COLLAPSING 
PEESSUEE  BEINa  450   POUNDS   PEB   SQUARE  INCH. 


Diameter  of 
Flue 
in  inches. 

Thickness  of  plates  in  decimal  parts  of  an  inch. 

For  a  Fine  10  feet 
long. 

For  a  Flue  20  feet 
long. 

For  a  Flue  80  feet 
long. 

12 
18 
24 
80 
86 
42 
48 

•291 
•350 
•899 
•442 
•480 
•51« 
•548 

•899 
•480 
•548 
•607 
•659 
•707 
•752 

•480 
•578 
•659 
•780 
•794 
•851 
•905 

AS   APPLICABLE   TO   LOCOMOTIVES.  329 


T 

If  p  =  806300   --  ,  then  by  transformation 


T2-l9=_tlli1L  and 
806300 

2-19  /  P  L  D 

T  = 


806300' 


If  now  we  put  p  the  collapsing  pressure  =  460  Ibs.,  L  =  10 
feet,  and  D  =  12  inches,  the  expression  becomes 


T  - 


_ log. -06734 


806300  2-19 

In  like  manner  the  quantities  L  and  D  can  easily  be  derived 
from  the  formula,  and  in  fact  the  equations  representing  them 
will  be 

806300  T2-19 


and 


PD 

806300  T2-19 


It  is  unnecessary  to  put  these  equations  into  words,  as  the 
rule  for  finding  the  collapsing  pressure  of  flues  is  not  much  re- 
quired, seeing  that  in  the  case  of  all  large  internal  flues  they  may 
be  strengthened  by  hoops  of  T  iron,  so  as  to  be  as  strong  as  the 
shell. 

PRACTICAL  EXAMPLE   OF   A  LOOOMOTIYB  BOILEB. 

It  will  be  useful  to  compare  the  results  given  by  these  com- 
putations with  the  actual  proportions  of  a  locomotive  boiler  of 
good  construction,  and  I  shall  select  as  the  example  one  of  the 
outside-cylinder  tank  engines  constructed  by  Messrs.  Sharp  and 
Co.  for  the  North-Western  Railway.  The  diameter  of  cylinder  in 
this  locomotive  is  15  inches,  and  the  length  of  the  stroke  20  inches. 
The  pressure  of  the  steam  in  the  boiler  is  80  Ibs.  per  square  inch. 
The  barrel  of  the  boiler  is  3  feet  6  inches  diameter,  and  10  feet 
8^  inches  long,  and  it  is  formed  of  iron  plates  fths  thick.  The 
junction  of  the  plates  is  effected  by  a  riveted  jump-joint,  which 
is  equal  in  strength  to  a  single  riveted-joint.  The  rivets  are  } 


330  PROPORTIONS    OF    STEAM-BOILERS, 

inch  in  diameter.  The  external  fire-box  is  of  iron  fths  thick, 
and  the  internal  fire-box  is  TVths  thick,  except  the  part  of  the 
tube-plate  where  the  tubes  pass  through,  which  is  f  inch  thick. 
The  internal  and  external  fire-boxes  are  stayed  together  by  means 
of  copper  stay-bolts,  £  inch  in  diameter,  and  pitched  4  inches 
apart.  The  roof  of  the  fire-box  is  supported  by  means  of  seven 
wrought-iron  ribs  If  inches  thick  and  3f  inches  deep,  which  rest 
at  the  ends  on  the  sides  of  the  fire-box,  while  the  fire-box  crown, 
being  bolted  to  the  ribs,  is  kept  up.  The  ribs  are  widened  out 
at  the  bolt-holes,  and  are  also  made  somewhat  deeper  there,  so 
that  only  a  surface  of  about  -J  inch  round  each  bolt  bears  on  the 
boiler  crown,  to  which  it  is  fitted  steam-tight.  To  assist  in  keep- 
ing up  the  crown,  the  cross-ribs  are  also  connected  with  the 
roof  of  the  external  fire-box.  The  water  space  left  between  the 
outside  and  inside  fire-box  is  about  3  inches,  and  the  inside  fire- 
box should  always  be  made  pyramidical,  to  facilitate  the  disen- 
gagement of  the  steam  from  the  surface  of  the  metal.  There  is 
a  glass  tube  and  three  gauge-cocks,  for  ascertaining  the  level  of 
the  water  in  the  boiler.  The  lowest  gauge-cock  is  set  3  inches 
above  the  roof  of  the  internal  fire-box,  the  next  3  inches  above 
that,  and  the  next  3  inches  above  that,  so  that  the  highest  cock 
is  9  inches  above  the  top  of  the  internal  fire-box. 

There  is  a  lead  plug  fths  of  an  inch  diameter  screwed  into 
the  top  of  the  fire-box.  But  the  usual  course  now  is  to  place  the 
lead  plug  in  a  cupped  brass  plug  rising  a  little  way  above  the 
furnace  crown,  so  that  the  lead  may  melt  before  the  plating  of 
the  crown  gets  red-hot,  should  the  supply  of  water  be  from  any 
cause  intercepted. 

The  boiler  is  fitted  with  159  brass  tubes,  10  feet  Tf  inches 
long,  If  inches  external  diameter,  and  -j^th  of  an  inch  thick, 
fixed  in  with  ferules  only  at  the  fire-box  end.  Such  tubes  last 
from  four  to  five  years,  and  they  are  now  made  thickest  at  the 
fire-box  end,  where  the  wear  is  greatest.  The  part  of  the  boiler 
above  the  tubes  is  supported  by  eight  longitudinal  stays,  running 
from  end  to  end  of  the  boiler.  The  back  tube-plate  is  of  iron 
fths  of  an  inch  thick.  The  smoke-box  is  J  inch  thick,  and  the 
chimney,  which  is  15  inches  diameter  at  bottom  and  12£  inches 


AS   APPLICABLE  TO   LOCOMOTIVES.  331 

at  top,  and  rises  13  feet  3  inches  above  the  rails,  is  jth  of  an  inch 
thick.  The  damper  for  regulating  the  draught  is  placed  at  the 
front  of  the  ash-pan,  and  there  is  another  similar  damper  at  tho 
hack  of  the  ash-pan  to  he  used  when  the  engine  is  made  to  travel 
backward,  which  tank  engines  can  the  better  do,  as  they  have 
no  tender.  The  surface  of  the  fire-grate  is  lOfths  square  feet. 
The  steam  ports  for  admitting  the  steam  to  the  cylinder  are  11 
inches  by  If  ths,  and  consequently  each  has  an  area  of  17'875  square 
inches.  The  branch  steampipe  leading  to  each  cylinder  has  J 
less  area  than  this.  The  blast-pipe  is  6f  inches  diameter,  taper- 
ing to  5  J  inches  diameter  at  the  top,  and  within  it  is  a  movable 
piece  of  taper  pipe,  which  may  be  raised  up  when  it  is  desired 
to  contract  the  blast  orifice.  The  consumption  of  coke  in  these 
engines  is  25  Ibs.  per  mile.  The  evaporation  in  locomotive 
boilers  is  7£  to  8  Ibs.  of  water  per  Ib.  of  coke,  and  in  locomotive 
boilers  working  without  expansion  the  evaporation  of  a  cubic 
foot  of  water  in  the  hour  will  be  about  equivalent  to  an  actual 
horse-power.  Now  if  the  speed  be  supposed  to  be  30  miles  an 
hour,  a  mile  will  be  performed  in  two  minutes ;  and  as  the  con- 
sumption per  two  minutes  is  25  Ibs.,  the  consumption  per  one 
minute  will  be  the  half  of  25  Ibs.,  or  say  12  Ibs.  per  minute ;  and 
the  consumption  in  60  minutes,  or  one  hour,  will  be  conse- 
quently 720  Ibs.  of  coke ;  and  if  8  Ibs.  of  water  are  evaporated 
by  1  Ib.  of  coke,  the  water  evaporated  per  hour  will  be  8  times 
720,  or  5760  Ibs.  Now  if  we  take  a  cubic  foot  of  water  at 
62-J-  Ibs.,  and  as  the  evaporation  of  a  cubic  foot  in  the  hour  is 
equivalent  to  a  horse-power,  5760  divided  by  62-J-  =  92,  will  be 
the  number  of  actual  horse-power  exerted  by  this  engine  under 
the  circumstances  supposed. 

Practically,  however,  locomotives  of  this  class  are  capable 
of  exerting  much  more  than  92  actual  horse-power;  for  all 
modern  locomotives  work,  to  a  certain  extent,  expansively, 
whereby  a  given  bulk  of  water  raised  into  steam  is  enabled  to 
exert  more  power,  and  further,  the  consumption  of  coal  per 
mile  may  be  increased  beyond  25  Ibs.,  with  a  corresponding  in- 
crease of  the  power  generated.  In  all  boilers,  indeed,  whether 
land,  marine,  or  locomotive,  the  evaporative  power  will  be 


332  PROPORTIONS   OF   STEAM-BOILERS. 

greatly  increased  by  every  expedient  which  increases  the  velocity 
of  the  draft,  and  if  arrangements  be  simultaneously  made  for  in- 
creasing the  temperature  of  the  furnace,  by  contracting  the 
escaping  orifice  over  the  bridge  or  through  the  flues,  the  expen- 
diture of  fuel  to  accomplish  any  given  evaporation  will  not  be 
increased.  In  this  way  marine  boilers  have  been  constructed 
with  only  12  square  feet  of  heating  surface  per  nominal  horse 
power,  and  in  which  the  consumption  was  only  2*  Ibs.  of  coal 
per  actual  horse  power,  as  will  be  seen  by  a  reference  to  page  52 
of  the  Introduction  to  my  '  Catechism  of  the  Steam  Engine.' 


CHAPTER  YI. 

POWER  AND  PERFORMANCE  OF  ENGINES. 

THE  manner  of  determining  the  nominal  power  of  an  engine 
has  heen  already  explained,  and  it  now  remains  to  show  in  what 
manner  its  actual  or  indicator  horse-power  may  be  determined. 

Construction  of  the  Indicator. — The  common  form  of  indica- 
tor applicable  to  engines  moving  at  low  rates  of  speed  I  have  al- 
ready described  in  my  '  Catechism  of  the  Steam-Engine.'  But 
in  the  case  of  engines  moving  at  high  rates  of  speed,  and,  in  fact, 
in  the  case  of  all  engines  to  which  the  steam  is  quickly  admitted, 
the  diagrams  formed  by  this  species  of  indicator  are  much  dis- 
torted, and  the  accuracy  of  the  result  impaired,  by  the  momen- 
tum of  the  piston  of  the  indicator  itself,  which  is  shot  up  sud- 
denly by  the  steam  to  a  point  considerably  higher  than  what 
answers  to  the  actual  pressure.  The  recoil  of  the  spring  again 
sends  the  piston  below  the  point  which  properly  represents  the 
pressure ;  and  in  interpreting  the  diagram  the  true  curve  is  sup- 
posed to  run  midway  between  the  crests  and  hollows  of  the 
waving  line  produced  by  these  oscillations.  Latterly  an  im- 
proved form  of  indicator,  called  Richards'  indicator,  has  been 
introduced,  which  is  represented  in  fig.  5,  of  which  the  main  pe- 
culiarity is  that  its  piston  is  very  light  and  has  a  very  small 
amount  of  motion,  so  that  its  momentum  is  not  sufficiently  great 
to  disturb  the  natural  line  of  the  diagram.  The  motion  of  the 
piston  of  the  indicator  is  multiplied  sufficiently  to  give  a  diagram 
of  the  usual  height  by  means  of  a  small  lever  jointed  to  the  top 
of  the  piston  rod.  To  the  end  of  this  lever  a  small  link,  carry- 


334 


RICHARDS'  INDICATOR. 
Fig.  5. 


RICHARDS'  INDICATOR.    (By  Elliot  Brothers,  Strand.) 


METHOD    OF   APPLYING  THE   INDICATOR.  335 

ing  the  pencil,  is  attached,  and  from  the  lower  end  of  this  small 
link  a  small  steel  radius  bar  proceeds  to  a  fixed  centre  on  a  suit- 
able part  of  the  instrument,  so  as  to  form  a  parallel  motion 
whereby  the  pencil  is  constrained  to  move  up  or  down  in  a  ver- 
tical direction.  The  paper  is  placed  upon  the  drum,  shown  in 
the  figure  with  a  graduated  scale,  and  the  string  causing  this 
drum  to  turn  round  and  back  again  on  its  axis  is  put  into  con- 
nection with  some  part  partaking  of  the  motion  of  the  piston  in 
the  usual  manner.  To  withdraw  the  pencil  from  the  paper,  the 
whole  parallel  motion  and  the  arms  carrying  it  are  turned  round 
upon  the  cylinder,  and  the  pencil  is  thus  made  readily  accessible. 
The  action  of  this  indicator  is  precisely  the  same  as  that  of  the 
common  indicator,  which,  having  been  described  in  my  '  Cate- 
chism of  the  Steam-Engine,'  need  not  be  further  noticed  here. 
But  in  this  indicator,  as  the  spring  is  very  stiff,  and  the  travel  of 
the  piston  correspondingly  small,  there  are  no  inconvenient  os- 
cillations of  the  pencil  such  as  occur  when  a  long  and  slender 
spring  is  employed. 

Method  of  applying  the  Indicator. — The  drum  being  put  into 
communication  with  some  part  of  the  engine  possessing  the  same 
motion  as  the  piston,  but  sufficiently  reduced  in  amount  to  be 
suitable  for  the  small  size  of  the  instrument,  the  drum  will  begin 
to  be  turned  round  when  the  piston  begins  its  forward  stroke ; 
and  the  string  having  drawn  it  round  in  opposition  to  the  ten- 
sion of  the  spring  coiled  at  the  bottom  of  it,  it  wiU*follow  that 
when  the  string  is  relaxed,  as  it  will  be  on  the  return  stroke  of 
the  piston,  the  drum  will  turn  back  again  to  its  original  position, 
and  its  motion  and  that  of  the  string  will  be  an  exact  miniature 
of  the  motion  of  the  piston.  The  pencil,  if  now  suffered  to  press 
against  the  paper,  will  describe  a  straight  line.  But  if  the  cock 
which  connects  the  cylinder  of  the  indicator  with  the  cylinder 
of  the  engine  be  now  opened,  the  pencil  will  no  longer  trace  a 
straight  line,  but  being  pressed  upward  during  the  forward 
stroke  by  the  steam,  and  being  sucked  downward  by  the  vacuum 
during  the  return  stroke,  if  the  engine  is  a  condensing  one,  or 
being  pressed  downward  by  the  spring  when  the  pressure  of  the 
steam  is  withdrawn,  as  it  will  bo  during  the  return  stroke,  it  is 


336 


POWER  AND   PERFORMANCE   OF   ENGINES. 


quite  clear  that  the  pencil  must  now  describe  a  figure  containing 
a  space  or  area,  and  the  figure  is  what  is  called  the  indicator  di- 
agram, and  the  amount  of  the  space  is  the  measure  of  the  amount 
of  the  power  exerted  at  each  stroke  by  the  engine.  This  will  be 
more  clearly  understood  by  a  reference  to  fig.  6,  which  is  an  in- 
dicator diagram  taken  from  a  steam  fire-engine  constructed  by 
Messrs  Shand,  Mason  and  Co.,  with  two  high-pressure  engines 
of  6J  inch  cylinders  and  7  inches  stroke,  with  a  pressure  on  the 


Ion                   <^  <(%{%  Forward  stroke.                 Admission          .£8-9 
r.                                        Steam  line.                         corner.          ,        . 

!    130- 

r 

*^ 

120 
110- 

100- 

90- 
90- 

70- 

o 

0 

o 

o 

if> 

o 

0 

O 

\f> 

0 

60 

o 

0 

Is 

o 

a 

N 

N 

§ 

N 

5 

01 

w 

0 

50- 

40- 

/ 

30- 

v 

/    20: 

\ 

"~    1 

EDUCATION 

LIME 

/            10. 

Hri3  L_J,  ..: 

L  _.._..  

L  

Atmospheric  Hne 
lieturn  stroke. 


Ciompressive 
corner. 


DIAGRAM  ILLUSTRATIVE  OF  THE  MODE  OP  COMPUTING  THE  HORSE-POWER. 

boiler  of  145  Ibs.  per  square  inch,  and  making  156  revolutions 
per  minute.  The  total  weight  of  this  engine  is  24  cwt.  2  qrs., 
and  by  a  reference  to  the  diagram  it  will  be  seen  that  the  mean 
pressure  urging  the  piston  is  117'5  Ibs.  per  square  inch,  which 
mean  pressure  is  ascertained  by  adding  together  the  pressure  at 
each  division  or  ordinate,  and  dividing  by  the  number  of  ordi- 
nates,  which  in  this  case  is  10.  The  mean  pressure  multiplied 
by  the  areas  of  the  cylinders  and  by  the  speed  of  the  piston  in 
feet  per  minute,  and  divided  by  33000  Ibs.,  gives  18'3  horses  as 


INTERPRETATION    OP   INDICATOR   DIAGRAMS.  337 

the  power  actually  exerted  by  this  engine.  The  weight  of  the 
engine  is  consequently  only  1-3  cwt.  per  actual  horse-power. 

The  advantage  of  taking  10  ordinates  instead  of  8  or  9  or  11 
is,  that  the  division  by  10  is  accomplished  by  merely  shifting  the 
position  of  the  decimal  point;  while  10  ordinates  are  enough  to 
enable  the  area  to  be  measured  accurately  enough  for  all  practi- 
cal purposes.  Thus  the  total  amount  of  the  pressures  in  the  di- 
agram, fig.  6,  taken  at  10  places,  is  1175  Ibs.,  and  the  tenth  of 
this,  or  11T5  Ibs.  per  square  inch,  is  the  mean  pressure  on  the 
piston  throughout  the  stroke.  It  is  clear  that  when  we  have 
got  the  mean  pressure  on  each  square  inch  of  the  piston,  we 
have  only  to  ascertain  the  number  of  square  inches  in  it,  and 
the  distance  through  which  it  moves  in  a  minute,  to  determine 
the  power,  and  the  indicator  enables  us  to  determine  the  mean 
pressure  on  the  piston  throughout  the  stroke  in  the  manner  just 
explained.  The  indicator  is  sometimes  applied  to  the  air-pump 
and  to  the  hot  well,  to  determine  the  varying  pressures  within 
them  at  different  parts  of  the  stroke ;  and  it  is  virtually  the 
stethoscope  of  the  engine,  as  it  enables  us  to  tell  whether  all  its 
internal  motions  and  pulsations  are  properly  performed. 

Mode  of  reading  Indicator  Diagrams. — In  the  preceding  di- 
agram the  piston  moves  in  the  forward  stroke  in  the  direction 
shown  by  the  arrow,  and  backward  on  the  return  stroke  in  the  di- 
rection shown  by  the  arrow.  In  all  diagrams  the  top  indicates  the 
highest  pressure,  and  the  bottom  the  lowest  pressure.  But  it  is 
quite  indifferent  whether  the  diagram  is  a  right-hand  or  left- 
hand  diagram ;  and  where  two  diagrams  are  shown  on  the  same 
piece  of  paper,  as  is  often  done,  that  which  represents  the  per- 
formance of  one  end  of  the  cylinder  is  generally  right-hand,  and 
that  which  represents  the  performance  of  the  other  end  of  the 
cylinder  is  generally  left-hand.  This  arrangement,  however,  is 
quite  immaterial,  that  which  alone  determines  the  power  exert- 
ed being  with  any  given  scale  the  area  shut  within  the  diagram. 

In  fig.  6,  the  steam  being  supposed  to  be  let  in  upon  the  pis- 
ton of  the  engine,  presses  the  piston  of  the  indicator  up  to  the 
point  shown  at  the  'admission  corner,'  and  as  the  piston  moves 
forward  the  steam  continues  to  press  upon  it  with  undiminished 
15 


338     POWER  AND  PERFORMANCE  OF  ENGINES. 

pressure,  until  close  to  the  end  of  the  stroke,  at  the  '  eduction 
corner,'  the  eduction  passage  is  opened ;  and  as  the  steam  con- 
sequently escapes  into  the  atmosphere  there  is  no  longer  the 
same  pressure  on  the  spring  of  the  indicator  as  hefore,  and  its 
piston  consequently  descends.  As,  however,  the  steam  cannot 
instantaneously  get  away,  the  pressure  does  not  descend  quite  so 
low  as  the  atmospheric  line.  The  eduction  passage,  it  appears 
by  the  diagram,  begins  to  be  opened  when  about  nine-tenths  of 
the  forward  stroke  has  been  completed,  and  it  also  begins  to  be 
shut  when  about  nine-tenths  of  the  return  stroke  has  been  corn- 
Fig.  7. 


DIAGRAM  TAKEN  FROM  STEAMER  '  ISLAND  QUEEN.' 

pleted,  as  appears  by  a  reference  to  the  '  compression '  corner, 
which  shows  that  the  back  pressure  begins  to  rise  before  the 
termination  of  the  stroke.  The  area  comprehended  between  the 
atmospheric  line  and  the  bottom  of  the  diagram  shows  the 
amount  of  back  pressure  resisting  the  piston,  which  in  this  dia- 
gram is  of  the  average  amount  of  5*1  Ibs. ;  and  this  increased 
back  pressure  at  the  '  compression  corner '  is*  produced  by  the 
compression  of  the  steam  shut  within  the  cylinder,  which  is  ac- 
complished by  the  piston  as  it  approaches  the  end  of  its  stroke. 
Various  examples  of  Indicator  Diagrams. — In  the  engine  of 
which  the  diagram  is  given  in  fig.  6,  the  steam  works  with  very 
little  expansion ;  but  in  fig.  7  we  have  a  diagram  taken  from  the 
steamer  'Island  Queen,'  which  shows  a  large  amount  of  expan- 


INTERPRETATION    OF   INDICATOR   DIAGRAMS.  339 

sion.  This  diagram  is  a  left-hand  diagram,  the  former  one, 
shown  in  fig.  6,  being  a  right-hand  diagram.  A  is  the  admission 
corner,  and  the  steam  is  only  admitted  until  the  piston  reaches 
the  position  answering  to  that  of  a  vertical  line  drawn  through 
a,  and  which  is  about  one-eighth  of  the  stroke.  The  steam  be- 
ing shut  off  from  the  cylinder  at  a,  thereafter  expands  until  the 
end  of  the  stroke  is  nearly  reached,  when  the  eduction  passage 
is  opened,  and  the  pencil  then  subsides  to  the  point  B,  at  which 
point  the  piston  begins  to  return.  The  straight  line  drawn 
across  the  middle  of  the  diagram  is  the  atmospheric  line ;  and  it 
is  traced  by  the  pencil  before  the  cock  of  the  indicator  commu- 
nicating with  the  cylinder  is  opened.  The  distance  of  the  line 
B  o  below  the  atmospheric  line  shows  the  amount  of  vacuum  ob- 
tained in  the  cylinder,  and  the  height  of  A  a  above  the  atmos- 
pheric line  shows  the  pressure  of  the  steam  subsisting  in  the 
cylinder.  This  diagram,  which  is  a  very  good  one,  is  obtained 
with  the  aid  of  a  separate  expansion  valve.  The  pressure  of  the 
steam  was  22  Ibs.  per  square  inch,  the  vacuum  14J  Ibs.,  and  the 
number  of  revolutions  per  minute  17. 

In  some  high-pressure  engines,  where  the  steam  is  allowed  to 
escape  suddenly  through  large  ports,  and  a  large  and  straight 
pipe,  there  is  not  only  no  back  pressure  on  the  piston,  but  a 
partial  vacuum  is  created  within  the  cylinder  by  the  momen- 
tum of  the  escaping  steam.  In  ordinary  condensing  engines 
the  momentum  of  the  steam  escaping  into  the  condenser  might 
in  some  cases  be  made  to  force  the  feed-water  into  the  boiler,  in 
the  same  manner  as  is  done  by  a  Giffard's  injector,  which  is  an 
instrument  that  forces  water  into  a  boiler  by  means  of  a  jet  of 
steam  escaping  from  the  same  boiler.  This  instrument  will  not 
act  if  the  temperature  of  the  feed-water  be  above  120°  Fahr.,  as 
in  such  case  the  steam  will  not  be  condensed  with  the  required 
rapidity.  As  the  steam  is  water  in  a  state  of  great  subdivision, 
and  as  the  particles  of  this  water  are  moved  with  the  velocity 
of  the  issuing  steam,  which  is  very  great,  we  have  in  effect  a 
very  small  jet  of  water  issuing  with  a  very  great  velocity,  and 
this  small  stream  would  consequently  balance  a  very  high  head 
of  water,  or,  what  comes  to  the  same  thing,  a  very  great  pres- 


340 


POWER  AND  PERFORMANCE  OF  ENGINES. 


sure.  Precisely  the  same  action  takes  place  when  the  steam  es- 
capes to  the  condenser ;  and  under  suitable  arrangements  the 
boiler  might  be  fed  by  aid  of  the  power  resident  in  the  educting 
steam,  and  indeed  the  function  of  the  air-pump  might  also  be 
performed  by  the  same  agency. 

In  fig.  8  we  have  an  example  of  the  diagrams  taken  from  the 
top  and  the  bottom  of  the  cylinder  disposed  on  the  same  piece 
of  paper,  those  on  the  left-hand  side  being  taken  from  the  top 
of  the  cylinder,  and  those  on  the  right-hand  side  being  taken 


DIAGRAMS  TAKEN  AT  MOORINGS  FROM  HOLYHEAD  PADDLE-STEAMER 

'  MUNSTER.' 

from  the  bottom  of  the  cylinder.  There  are  three  diagrams 
taken  from  each  end  with  different  degrees  of  expansion.  A  is 
the  admission  corner  of  the  three  diagrams,  taken  from  the  top 
of  the  cylinder,  and  a  a  a  are  the  three  several  points  at  which 
the  steam  is  cut  off  in  these  three  diagrams.  Thereafter  the 
steam  continues  to  expand,  and  the  pressure  gradually  to  fall, 
until  the  points  5  I  5  are  reached,  when  the  eduction  passage  is 
opened  to  the  condenser,  and  the  pressure  then  falls  suddenly  to 
the  point  B.  The  line  B  B'  represents  the  amount  of  exhaustion 


DIAGRAMS  OF  HOLYHEAD  STEAMER. 


341 


attained  within  the  cylinder  measured  downward  from  the  at- 
mospheric line  M  L  ;  and  ccc  represent  the  three  points  at  which 
compression  begins,  answering  to  the  three  degrees  of  expansion. 
The  letters  A',  a',  &',  B',  and  c'  represent  the  corresponding  points 
for  each  of  the  three  diagrams  taken  from  the  bottom  of  the  cyl- 
inder ;  and  the  amount  of  correspondence  in  the  right-hand  and 
left-hand  diagrams  shows  the  amount  of  accuracy  with  which 
the  valves  are  set  to  get  a  similar  action  at  each  end  of  the  cyl- 

Fig.  9. 


DIAGRAMS  TAKEN  FROM  HOLYHEAD  PADDLE-STEAMER  '  ULSTER*  WHBN 
UNDER  WAY. 

inder.  The  diagrams  given  above  were  taken  from  the  Holy- 
head  steam-packet '  Munster,'  the  engines  of  which  were  con- 
structed by  Messrs.  Boulton  and  Watt.  The  cylinders  are  oscil- 
lating, of  96  inches  diameter  and  T  feet  stroke.  The  pressure  of 
steam  was  2616  Ibs.  per  square  inch,  vacuum  25£  Ibs.,  and  the 
number  of  strokes  per  minute  9 — the  vessel  having  been  at  moor- 
ings at  the  time.  It  will  be  seen  by  these  diagrams  that  the  amount 
of  lead  upon  the  eduction  side,  or  the  equivalent  distance  which 
the  piston  is  still  from  the  end  of  the  stroke  when  eduction  begins 
to  take  place,  corresponds  in  every  instance  with  the  amount  of 


842      POWER  AND  PERFORMANCE  OF  ENGINES. 

the  compression,  since,  in  fact,  by  shifting  the  eccentric  round 
to  let  the  steam  out  of  the  cylinder  before  the  end  of  the  stroke, 
the  valve  will  be  equally  shifted  to  shut  the  educting  orifice  be- 
fore the  end  of  the  stroke,  and  thus  to  keep  within  the  cylinder 
any  vapour  left  in  it  when  the  valve  has  been  shut,  and  which 
is  thereafter  compressed  by  the  piston  until  the  end  of  the  stroke 
is  reached,  or  until  the  valve  opens  the  communication  with  the 
boiler. 

Fig.  9  represents  a  diagram  taken  from  the  top,  and  another 
taken  from  the  bottom  of  one  of  the  cylinders  of  the  Holyhead 
paddle-steamer  'Ulster,'  a  vessel  of  the  same  power  and  dimen- 
sions as  the  '  Munster,'  and  the  engines  also  by  Messrs.  Boulton 

Fig.  10. 


DIAGRAMS  FROM  STEAMER  '  ULSTER  '  AT  4  1-2  STROKES. 
(STEAM  THROTTLED  BY  THE  LINK.) 

and  "Watt.  When  these  diagrams  were  taken  the  pressure  of  the 
steam  in  the  boiler  was  26  Ibs.  per  square  inch,  the  vacuum  in 
the  condenser  13  Ibs.  per  square  inch,  and  the  engine  was  mak- 
ing 23  strokes  per  minute.  The  mean  pressure  on  the  pistons, 
obtained  by  taking  a  number  of  ordinates,  as  in  fig.  6,  reckoning 
up  the  collective  pressure  at  each,  and  dividing  by  the  number 
of  ordinates,  was  28*27  Ibs.  It  is  immaterial  what  number  of 
ordinates  is  taken,  except  that  the  more  there  are  taken  the  more 
accurate  will  be  the  result. 

In  fig.  10  we  have  diagrams  taken  from  top  and  bottom  in  the 
same  engine,  when  slowed  to  4J  strokes  per  minute,  partly  by 
closing  the  throttle  valve,  and  partly  by  shifting  the  link  towards 
its  mid-position.  In  these  diagrams  nearly  the  whole  areas  are 


DIAGRAMS    OF    HOLYHEAD    MAIL    STEAMER.  343 

below  the  atmospheric  line.  But  on  the  left-hand  corner  of  one 
of  the  figures  a  loop  is  formed,  which  often  appears  in  engines 
employing  the  link,  and  the  meaning  of  which  it  is  necessary  to 
explain.  The  extreme  point  of  the  diagram  in  every  instance 
answers  to  the  length  of  the  stroke;  and  if  the  steam  is  pent  up 
in  the  cylinder  hy  the  eduction  passage  being  shut  before  the 
end  of  the  stroke,  or  if  it  be  suffered  to  enter  from  the  boiler  be- 
fore the  stroke  is  ended,  the  pencil  will  be  pushed  up  to  its  high- 
est point  before  the  stroke  is  ended,  and  as  the  paper  still  con- 
tinues to  move  onward  the  upper  part  of  the  loop  is  formed.  If 
the  pressure  within  the  cylinder  when  the  piston  returns  were 
to  be  precisely  the  same  as  when  the  piston  advances  during  this 
part  of  its  course,  the  loop  would  be  narrowed  to  a  line.  But  as 
the  advance  of  the  piston  when  the  valve  is  very  little  opened 
somewhat  compresses  the  steam,  and  as  its  recession  when  the 
valve  is  very  little  opened  somewhat  wire-draws  it,  the  pressures 
while  the  piston  advances  and  retires  through  this  small  distance, 
although  the  cylinder  is  open  to  the  boiler  by  means  of  a  small 
orifice,  will  not  be  precisely  the  same ;  and  the  higher  pressure 
will  form  the  upper  part  of  the  loop,  and  the  lower  pressure  the 
lower  part.  In  fig.  10,  by  following  the  outline  of  the  left-hand 
diagram,  it  will  be  seen  that  the  steam  begins  to  be  compressed 
within  the  cylinder  when  about  three-fourths  of  the  stroke  has 
been  completed ;  and  the  pencil  consequently  begins  to  rise 
somewhat  above  its  lowest  point.  But  as  the  vapour  within  the 
cylinder  is  very  rare,  the  rise  is  very  little  until,  when  the  piston 
is  about  one-eighth  part  of  its  motion,  or  about  8  inches  from 
the  end  of  the  stroke,  the  steam-valve  is  slightly  opened,  when 
the  piston  of  the  indicator  is  compelled  to  ascend  to  the  point 
answering  to  the  pressure  within  the  cylinder  thus  produced. 
As  the  opening  from  the  boiler  continues,  and  the  piston  by  ad- 
vancing against  the  steam,  instead  of  receding  from  it,  compress- 
es rather  than  expands  the  steam  admitted  into  the  cylinder,  the 
pressure  continues  to  rise  somewhat  to  the  end  of  the  stroke ; 
when  the  piston  of  the  engine,  having  to  move  in  the  opposite 
direction,  the  steam  within  the  cylinder  will  be  expanded,  and 
any  still  entering  will  be  wire-drawn  in  the  contracted  passage, 


344 


POWER  AND  PERFORMANCE  OF  ENGINES. 


and  the  pressure  will  fall.  Under  such  circumstances  a  loop  will 
necessarily  be  formed  at  the  corner  of  the  diagram,  such  as  is 
shown  to  exist  at  the  left-hand  corner  of  fig.  10.  The  reason 
why  there  is  no  corresponding  loop  at  the  right-hand  corner  of 
the  right-hand  diagram  is  simply  because  the  valve  is  somewhat 
differently  set  at  one  end  of  the  engine  from  what  it  is  at  the 
other ;  and  the  angles  of  the  eccentric  rods  will  generally  cause 

Fig.  11. 


DIAGEAM  FROM  AIR-PUMP  OF  STEAMER  '  ULSTER.' 
(19  BErOLTTTIONS  PEB  MINUTE.) 

some  small  difference  in  the  action  of  the  valve  at  the  different 
ends  of  the  engine. 

Diagrams  from  the  Air-Pump. — Fig.  11  is  a  diagram  taken 
from  the  air-pump  of  the  '  Ulster,'  when  the  engine  was  making 
19  revolutions  per  minute.  In  this  diagram  the  pencil  begins  to 
ascend  from  that  point  which  marks  the  amount  of  exhaustion 
existing  in  the  air-pump,  and  it  rises  very  slowly  until  about 
two-thirds  of  the  stroke  of  the  pump  has  been  performed,  when 
it  shoots  rapidly  upwards,  indicating  that  at  this  point  the  water 
is  encountered  which  has  to  be  expelled.  Midway  between  the 


DIAGRAMS   TAKEN    FROM   THE    AIR-PUMP.  345 

atmospheric  line  and  the  highest  point  of  ascent,  the  delivery 
valve  begins  to  open,  and  somewhat  relieves  the  pressure ;  and 
there  is  consequently  a  wave  in  the  diagram  on  that  point.  But 
the  inertia  of  the  water  in  the  hot-well  has  then  to  be  encoun- 
tered, and  an  amount  of  pressure  is  required  to  overcome  this 
inertia,  which  is  measured  by  the  highest  point  to  which  the 
pencil  ascends.  So  soon  as  the  water  in  the  hot-well  and  waste- 
water  pipe  has  been  put  into  motion,  the  motion  is  continued 
by  its  own  momentum,  without  a  sustained  pressure  being  re- 
quired to  be  exerted  by  the  bucket  of  the  pump  ;  and  the  pres- 
sure in  the  pump  consequently  falls,  as  is  shown  by  the  descent 
of  the  piston  of  the  indicator  towards  the  end  of  the  stroke. 

Fig.  12. 


DIAGRAM  FEOM  AIR-PUMP  OF  STEAMER  '  ULSTER. 
(7  STBOKE8  PEB  MINUTE.) 

The  effect  of  partially  closing  the  throttle-valve  of  an  engine 
so  as  to  diminish  the  speed,  will  be  to  reduce  the  momentum  of 
the  water  in  the  hot-well,  and  correspondingly  to  reduce  the 
maximum  pressure  which  the  pump  has  to  exert.  But  the  ef- 
fect will  also  be  to  fill  the  pump  with  water  through  a  larger 
proportion  of  its  stroke ;  and  if  the  engine  were  to  be  slowed 
very  much  by  shutting  off  the  steam,  without  correspondingly 
shutting  off  the  injection,  the  air-pump  at  its  reduced  speed 
would  be  unable  to  deliver  all  the  water,  which  would  conse- 
quently overflow  into  the  cylinder  and  probably  break  down  the 
engine.  In  fig.  12  we  have  an  air-pump  diagram  taken  from  the 
steamer  'Ulster,'  when  the  speed  of  the  engine  was  reduced  to 


346  POWER   AND   PERFORMANCE    OF   ENGINES. 

six  strokes  per  minute ;  and  it  will  be  observed  that  we  have 
no  longer  the  same  amount  of  maximum  pressure  in  the  pump, 
nor  the  same  sudden  fluctuations.  The  pump,  however,  is  filled 
for  a  greater  proportion  of  its  stroke ;  and  the  maximum  pres- 
sure once  created,. is  constant,  and  does  not  rise  much  above  the 
pressure  of  the  atmosphere,  being,  in  fact,  the  simple  pressure 
due  to  the  pressure  of  the  atmosphere,  and  that  of  the  column 
of  water  intervening  between  the  level  of  the  air-pump  and  that 
of  the  waste-water  pipe. 

Diagram  illustrative  of  the  evils  of  Small  Ports. — Fig.  13  is 
a  diagram  taken  from  a  pumping-engine  in  the  St.  Katherine's 
Docks,  and  is  introduced  mainly  to  show  the  detrimental  effect 

Fig.  13. 


DIAGRAM  TAKEN  FROM  PUMPING-ENGINE,  ST.  KATHERINE'S  DOCKS. 

of  an  insufficient  area  of  the  eduction  passages.  The  steam  is 
supposed  to  enter  at  the  left-hand  corner,  but  as  the  speed  of  the 
piston  accelerates,  as  it  does  towards  the  middle  of  the  stroke, 
the  pressure  falls,  from  the  port  being  small  and  the  steam  wire- 
drawn. Towards  the  other  end  of  the  stroke  the  pressure  would 
again  rise,  but  that  it  is  hindered  from  doing  so  by  the  condensa- 
tion within  the  cylinder,  which  is  considerable,  as  the  engine 
works  at  the  low  speed  of  12  strokes  per  minute,  lifting  the  wa- 
ter 9J  feet.  The  eduction  corner  of  the  diagram  is  very  much 
rounded  away,  from  the  inadequate  size  of  the  ports ;  and  the 
eduction  will  also  be  impeded  by  any  condensed  water  within 
the  cylinder,  which,  unless  got  rid  of  by  other  arrangements,  will 
have  to  be  put  into  motion  by  the  escaping  steam.  The  mean 


DIAGRAMS   TAKEN   FROM   THE   HOT-WELL.  347 

pressure  exerted  on  the  piston  of  this  engine  is  only  12 '45  Ibs. 
per  square  inch,  although  it  operates  without  expansion  ;  and  it 
may  be  taken  as  a  fair  example  of  eneligible  construction. 

Diagrams  showing  the  momentum  of  the  Indicator  piston. — 
Fig.  14  is  a  pair  of  diagrams  taken  from  one  of  the  engines  of 
H.  M.  S.  '  Orontes.'  This  vessel,  which  is  300  feet  1  inch  long, 
44  feet  8  inches  broad,  and  2,823  tons,  has  horizontal  direct 
acting  engines  of  500  horse-power,  constructed  by  Messrs  Boul- 

Fig.  14. 


DIAGRAM   TAKEN   FROM   H.M.    TROOP-STEAMER  '  ORONTES.' 

ton  and  "Watt.  With  a  midship  section  of  644  square  feet,  and  a 
displacement  of  3,400  tons,  the  vessel  attained  a  speed  on  her  offi- 
cial trial,  of  12-622  knots,  with  a  pressure  of  steam  in  the  boiler 
of  25  Ibs.  per  square  inch,  61  revolutions  per  minute,  the  engines 
exerting  2,249  horse-power.  On  one  occasion  the  speed  obtained 
was  13-3  knots.  With  an  area  of  immersed  section  of  Y81  square 
feet,  and  a  displacement  of  4249  tons,  the  speed  attained  was 
12*354  knots,  with  2,143  horse-power.  There  are  two  horizontal 


348  POWER   AND   PERFORMANCE    OF    ENGINES. 

engines  of  71  inches  diameter,  and  3  feet  stroke.  The  screw  is 
18  feet  diameter,  25  feet  pitch,  and  4  feet  long,  and  the  slip  of 
the  screw  was  found  to  vary  between  13  and  16  per  cent.  When 
the  diagrams  represented  in  fig.  14  were  taken,  the  pressure  of 
the  steam  in  the  boiler  was  2H  Ibs.  of  the  vacuum,  in  the  con- 
denser llf  Ibs.,  and  the  engine  was  making  60  revolutions  per 
minute.  If  ordinates  be  taken  in  the  case  of  these  diagrams,  and 
the  mean  pressure  be  thus  determined,  it  will  be  found  to  amount 
to  25 '22  Ibs.  per  square  inch.  In  these  diagrams  the  waving  line 
formed  by  the  pencil,  owing  to  the  momentum  of  the  piston  of 
the  indicator,  is  very  plainly  shown ;  and  although  such  irregu- 
larities will  not  materially  impair  the  accuracy  of  the  result,  if  a 
sufficient  number  of  ordinates  be  taken  correctly  to  measure  the 
irregularity,  yet  it  is  greatly  preferable  to  employ  an  indicator 
which  will  be  as  free  as  possible  from  the  disturbing  influence  of 
the  momentum  of  its  own  moving  parts.  In  this  engine,  as  in 
most  of  Messrs.  Boulton  and  Watt's  engines,  there  is  a  great 
similarity  in  the  diagrams  taken  from  each  end  of  the  cylinder 
— a  result  mainly  produced  by  giving  a  suitable  length  to  the 
eccentric  rods,  by  moving  up  or  down  the  links  vertically  by 
a  screw,  instead  of  by  a  lever  moving  in  the  arc  of  a  circle,  and 
placing  the  projecting  side  of  the  eccentric  suitably  with  the 
curvature  of  the  link,  since,  if  placed  in  one  position,  it  will  aggra- 
vate the  distortion  produced  by  the  angle  of  the  eccentric  rods, 
and  if  placed  in  the  opposite  position  it  will  correct  this  dis- 
tortion. 

Fig.  15  represents  a  series  of  diagrams  from  each  end  of  one 
of  the  engines  of  the  '  Orontes,'  formed  by  allowing  the  pencil  to 
rest  on  the  paper  during  many  revolutions,  instead  of  only  dur- 
ing one.  These  diagrams  show  small  differences  between  one 
another,  mainly  in  the  mean  pressure  of  the  steam. 

Fig.  16  represents  two  diagrams  taken  from  the  engines  of 
the  iron-clad  screw  steamer  'Eesearch,  fitted  with  horizontal 
engines,  with  50-inch  cylinders,  and  2  feet  stroke.  With  a  pres- 
sure of  steam  in  the  boiler  of  22  Ibs.,  and  with  a  vacuum  in  the 
condenser  of  12f  Ibs.  per  square  inch,  the  mean  pressure  on  the 
piston  shown  by  the  diagrams  is  24*55  Ibs.,  the  engine  making 


DIAGRAMS   FROM    DIFFERENT    STEAMERS. 


349 


85  revolutions  per  minute.  This  engine  is  fitted  with  surface 
condensers.  The  serrated  deviation  at  a  is  caused  by  the  mo- 
mentum of  the  piston  of  the  indicator. 

In  fig.  17  we  have  two  diagrams,  taken  from  opposite  ends 
of  one  of  the  engines  of  H.M.S.  '  Barossa.'  This  vessel  is  225 
feet  long,  40  feet  8  inches  broad,  and  1,702  tons  burden.  With 
a  mean  draught  of  water  15-J-  feet  or  thereabout,  the  area  of  mid- 
ship section  is  466  square  feet,  and  the  displacement  1,780  tons. 
The  vessel  is  propelled  by  two  horizontal  engines,  with  cylinders 

Fig.  15. 


DIAGRAMS  TAKEN  FEOM  SCREW  STEAMER  '  ORONTES.' 

of  64  inches  diameter  and  3  feet  stroke,  the  nominal  power 
being  400  horses.  On  the  official  trial  this  vessel  realised  a 
speed  of  11-92  knots,  with  a  pressure  of  steam  in  the  boiler  of 
20  Ibs.  per  square  inch,  and  with  an  indicated  power  of  1798'2 
horses,  the  engine  making  66  revolutions  per  minute.  The 
screw  is  16  feet  diameter,  24  feet  pitch,  and  3  feet  long,  and  the 
slip  at  the  time  of  trial  was  23-71  per  cent.  When  the  diagrams 
shown  in  fig.  17  were  taken,  the  pressure  of  steam  in  the  boiler 
was  19  Ibs.  per  square  inch  ;  vacuum  in  condenser  12£  Ibs.  per 
square  inch,  the  revolutions  66  per  minute,  and  the  mean  pres- 


350 


POWER  AND  PERFORMANCE  OF  ENGINES. 


sure  on  the  piston  22*3  Ibs.  per  square  inch.  The  area  of  a  cylin- 
der of  64  inches  diameter  is  3216-2  square  inches,  the  douhle  of 
which  (as  there  are  two  cylinders)  is  6433'8  square  inches,  and 
as  there  22'3  Ibs.  on  each  square  inch,  there  will  be  a  total  pres- 
sure of  6433-8  times  22-3,  or  143,473-74  Ibs.  urging  the  pistons, 
and  as  the  length  of  the  double  stroke  is  6  feet,  the  power  ex- 
erted will  be  equal  to  6  times  143,473-74  Ibs.,  or  860,840-44 

Fig.  16. 


INDICATOR  DIAGRAMS  PROM  IRON-CLAD  STEAMER  '  RESEARCH.' 

foot-pounds  per  stroke,  and  as  there  are  66  strokes  per  minute, 
there  will  be  66  times  this,  or  56,797,869*04  foot-pounds  exerted 
per  minute.  As  an  actual  horse-power  is  33,000  foot-pounds 
per  minute,  we  shall,  by  dividing  56,797,869-04  by  33,000,  get 
the  actual  power  exerted  by  this  engine  at  the  time  the  above 
diagrams  were  taken,  and  which,  by  performing  the  division,  we 
shall  find  to  be  1721-1  horses. 

Various  Diagrams. — Fig.  18  is  a  diagram  taken  from  the 
air-pump  of  the  '  Barossa,'  which  is  a  double-acting  pump.    The 


DIAGRAMS    FROM    DIFFERENT    STEAMERS. 


351 


injection  was  all  on  at  the  time  this  diagram  was  taken,  and  the 
vacuum  was  only  11  Ibs.  per  square  inch.  In  my  'Catechism 
of  the  Steam-Engine,'  published  in  1856, 1  drew  attention  to  the 
fact  of  the  existence  of  very  imperfect  vacuums  in  engines  with 

Fig.  17. 


DIAGRAMS  TAKEN  FROM  H.  M.  STEAMEE  '  BAROSSA.' 

double-acting  air-pumps,  the  buckets  of  which  move  at  a  high 
rate  of  speed ;  and  I  also  pointed  out  the  cause  of  this  imperfect 
vacuum,  which  I  showed  to  be  consequent  on  the  lodgment  of 


Fig.  18. 


tfcr-10 


AIR-PUMP  DIAGRAM  FROM  H.  1C.  STEAMER  '  BAROSSA.' 

large  quantities  of  water  between  the  foot  and  delivery-valves 
at  the  end  of  the  pump,  into  which  water  the  pump  forced  in 
the  air  or  drew  it  out  without  ejecting  it  from  the  pump  at  all.  I 
consequently  recommend  that  in  all  pumps  of  this  class  the  bucket 


352      POWER  AND  PERFORMANCE  OP  ENGINES. 

and  valve-chambers  should  be  so  contrived  that  every  particle 
of  water  would  be  forced  out  of  the  puinp  at  every  stroke.  But 
up  to  the  present  time  I  do  not  find  that  this  recommendation 
has  been  generally  adopted,  and  in  nearly  every  species  of  direct- 
acting  screw-engine  operating  by  a  jet  in  the  condenser,  the 
vacuum  is  much  worse  than  it  was  in  the  old  class  of  paddle- 
engines,  or  even  in  the  land  engines  made  by  Watt  nearly  a  cen- 
tury ago. 

In  fig.  19  we  have  an  example  of  diagrams  taken  from  the  top 
and  bottom  of  one  of  the  paddle-engines  of  the  steamer  '  Great 
Eastern,'  constructed  by  Messrs.  J.  Scott  Russell  and  Co.  These 
engines  are  oscillating  engines  of  74  inches  diameter  of  cylinder, 
and  14feet  stroke,  making  10  revolutions  per  minute,  and  there  are 

Fig.  19. 


DIAGRAMS  FROM   PADDLE-ENGINES   OP    '  GREAT  EASTERN.' 

four  cylinders,  or  two  to  each  wheel.  The  mean  pressure  on  the 
piston  which  these  diagrams  exhibit  is  22'2  Ibs.  per  square  inch, 
from  which,  with  the  other  particulars,  it  is  easy  to  compute  the 
power. 

In  fig.  20  we  have  two  different  pairs  of  diagrams.  The 
larger  pair  is  taken  from  one  of  the  engines  of  the  paddle- 
steamer  '  Ulster,'  and  the  smaller  pah1 — represented  iu  dotted 
lines — is  taken  from  the  engines  of  the  paddle-steamer  '  Victoria 
and  Albert.'  In  the  case  of  the  '  Ulster '  the  pressure  of  steam 
in  the  boiler  when  the  diagram  was  taken  was  26  Ibs.  per  square 
inch,  and  the  vacuum  in  the  condenser  13  Ibs.  per  square  inch. 
The  number  of  strokes  per  minute  was  23,  the  mean  pressure  on 
the  piston  28'77  Ibs.  per  square  inch,  and  indicated  horse-power 


DIAGRAMS    FROM   DIFFERENT   STEAMERS. 


353 


4,100.  The  '  Victoria  and  Albert '  has  two  oscillating  engines,  with 
88-inch  cylinders  and  7-feet  stroke.  The  pressure  of  the  steam 
in  the  hoilers  when  the  diagrams  were  taken  was  26  Ibs.  per 
square  inch  ;  of  the  vacuum  12£  Ibs.  per  square  inch  ;  the  mean 
pressure  on  the  piston  22-87  Ibs.  per  square  inch,  and  the  num- 
ber of  strokes  per  minute  25 '4.  The  area  of  an  88-inch  cylinder 
is  6082'!  square  inches,  and  the  area  of  two  such  cylinders  is  the 

Fig.  20. 


COMPARATIVE  DIAGRAMS  PROM  'ULSTER*  AND  'VICTORIA  AND  ALBERT.' 

double  of  this,  or  12,164*2  square  inches,  and  as  there  are  22*87 
Ibs.  on  each  square  inch,  the  total  pressure  urging  both  pistons 
will  be  12,164-2  times  22*87  or  278,195  Ibs.  Now,  as  the 
length  of  Ahe  stroke  is  7  feet,  and  as  the  piston  traverses  it  each 
way  in  each  revolution,  the  piston  will  travel  14  feet  for  each 
revolution,  and  278,195  multiplied  by  14  will  give  3,894,730  as 
the  number  of  foot-pounds  exerted  in  each  stroke ;  or,  as  there 
are  25*4  strokes  each  minute,  there  will  be  25 '4  times  3,894,- 


354  POWER   AND    PERFORMANCE    OF   ENGINES. 

730,  or  98,926,142  foot-pounds  exerted  each  minute.  Dividing 
this  by  33,000,  we  get  the  power  exerted  by  this  engine  as 
equal  to  2997'7  actual  horse-power. 

In  the  diagrams  of  the  '  Victoria  and  Albert,'  it  will  be  re- 
marked there  is  a  greater  disparity  in  the  period  of  the  admission 
of  the  steam  than  in  the  case  of  the  diagrams  of  the  '  Ulster,' 
arising  from  the  valves  not  being  so  accurately  set. 

Diagram  showing  wrong  setting  of  Valves. — In  fig.  21  are 
given  two  diagrams,  taken  from  an  engine  making  200  strokes 
per  minute,  applied  to  work  the  exhausting  apparatus  employed 
by  the  Pneumatic  Despatch  Company  to  shoot  letters  and  par- 
cels through  a  tube.  These  diagrams  show  that  the  valve  is 
wrongly  set,  and  that  at  one  end  of  the  cylinder  the  steam  is  ad- 
Fig.  21. 


DIAGRAMS  FEOM  ENGINE  OF  PNEUMATIC  DESPATCH  COMPANY1. 

mitted  too  soon,  and  at  the  other  end  too  late.  By  following  the 
right-hand  diagram  it  will  be  seen  that  the  eduction  passage  is 
closed  when  about  half  the  stroke  has  been  performed,  and  that 
the  steam  is  admitted  in  front  of  the  piston  when  about  one- 
fourth  of  the  stroke  has  still  to  be  performed,  whereas  the  left- 
hand  diagram  shows  that  a  considerable  part  of  the  stroke  has 
been  performed  before  that  end  of  the  cylinder  begins  to  get 
steam.  The  action  in  this  case  would  be  amended  by  shifting 
round  the  eccentric.  The  mean  pressure  on  the  piston  shown 
by  these  diagrams  is  only  10'79  Ibs.  per  square  inch.  % 

Diagram  showing  the  necessity  of  large  Ports  for  high  speeds 
of  Piston. — Fig.  22  represents  two  diagrams  taken  from  the 
same  engine  with  the  unequal  action  at  the  different  ends  of  the 
cylinder  corrected.  But  the  diagrams  show  that  the  engine  has 


DIAGRAMS    FROM    FAST   EXGINES. 


355 


not  enough  lead  in  the  valves,  and,  moreover,  that  the  passages 
are  too  small  for  the  speed  with  which  the  engine  works.  It 
would  be  an  advantage  to  increase  either  the  width  or  the 
amount  of  travel  of  the  valve  of  this  engine,  or  hoth;  as  also  to 
give  more  lead,  so  that  the  steam  would  be  able  to  attain  and 
maintain  its  proper  pressure  at  the  beginning  of  the  stroke,  and 
until  it  is  purposely  cut  off.  The  mean  pressure  of  steam  on  the 
piston  shown  by  the  diagrams  represented  in  fig.  22  is  13'36  Ibs. 
per  square  inch. 

Diagrams  illustrative  of  the  action  of  the  Link  Motion. — 
In  fig.  23  we  have  a  diagram  taken  from  a  horizontal  engine, 
with  27-inch  cylinder  and  3-feet  stroke,  constructed  by  Messrs. 
Boulton  and  Watt,  employed  to  work  the  Portsmouth  Floating 

Fig.  22. 


DIAGRAMS  FBOM  ENGINE  OP  PNEUMATIC  DESPATCH  COMPANY. 

Bridge.  The  steam  is  cut  off  by  the  link  so  as  to  make  the  ad- 
mission almost  the  least  possible,  so  as  to  test  the  engine  itself 
before  the  chains  which  draw  the  bridge  backward  and  forward 
had  been  applied.  With  the  steam  cut  off  thus  early  there  is 
necessarily  a  very  large  amount  of  expansion,  and  also  a  very 
large  amount  of  cushioning ;  and  it  will  be  observed  that  the 
steam  begins  to  be  compressed  at  not  much  less  than  half-stroke. 
With  this  amount  of  expansion  the  link  is  2£  inches  from  the 
centre.  The  pressure  of  steam  in  the  boiler  was  22  Ibs.,  and 
that  of  the  vacuum  in  the  condenser  11  Ibs.  per  square  inch, 
when  this  diagram  was  taken  ;  and  the  engines  ran  without  the 
chains  at  40  revolutions  per  minnte. 

Fig.  24  is  another  diagram  taken  from  the  same  engine  with 


356 


POWER  AND  PERFORMANCE  OF  ENGINES. 


the  link  in  the  same  place.  Pressure  of  steam  in  boiler,  21  Ibs. 
per  square  inch;  pressure  of  vacuum  in  condenser,  lljlbs.  per 
square  inch;  number  of  revolutions  per  minute,  35.  In  this 
diagram,  and  also  in  the  last,  we  have  a  small  loop  formed  at 
the  top  of  the  diagram,  from  causes  already  explained. 

In  fig.  25  we  have  another  diagram  taken  from  the  same  en- 
gine, but  in  this  case  the  steam  is  not  shut  off  by  the  link  but 
by  the  throttle-valve,  and  there  is  consequently  very  little 
cushioning,  and  the  loop  at  the  top  of  the  diagram  almost  dis- 

Fig.  23. 


DIAGRAM  FROM  ENGINE  OF  PORTSMOUTH  FLOATING  BRIDGE. 
(ENGINE  THROTTLED  BY 


appears.  When  the  diagram  was  taken  the  pressure  of  steam 
in  the  boiler  was  22  Ibs.,  and  of  the  vacuum  in  the  condenser 
11J-  Ibs.  per  square  inch,  and  the  number  of  revolutions  per 
minute  was  38. 

Figs.  26,  27,  and  28  are  diagrams  taken  by  Eichards'  indi- 
cator from  Allen's  engine,  in  the  United  States  department  of 
the  International  Exhibition  of  1862.  In  this  engine  the  diam- 
eter of  the  cylinder  was  8  inches  ;  length  of  stroke,  24  inches  ; 
pressure  of  steam  in  boiler,  49  Ibs.  per  square  inch;  revolutions 
per  minute,  150. 


DIAGRAMS   FROM   FAST   ENGINES. 


357 


Diagrams  illustrative  of  action  of  Air-pump  and  Hot-well. 
-Fig.  29  is  a  diagram  taken  from  the  air-pump  of  the  Duke  of 


Fig.  24. 


Us. 


DIAGRAM  FROM  EXRINK  OF  PORTSMOUTH  FLOATING  BRIDGE. 
(ENGINE  THROTTLED  BY  LINK.) 

Sutherland's  yacht '  Undine,'  a  vessel  fitted  with  two  inverted 
angular  engines,  with  cylinders  24  inches  diameter  and  15  inches 

Fig.  25. 


DIAGRAM  FROM  ENGINE  OF  PORTSMOUTH  FLOATING  BRIDGE. 
(ENGINE  THEOTTLED  BY  THROTTLE- VALVE.) 


358      POWER  AND  PERFORMANCE  OF  ENGINES. 

stroke.  When  this  diagram  was  taken  the  ordinary  amount  of 
injection  was  on,  and  the  engine  was  working  at  moorings  at 
72  strokes  per  minute.  There  was  also  an  air-vessel  on  the 
hot-well.  In  fig.  80  we  have  a  diagram  taken  from  the  air-pump 
of  the  same  engine,  with  an  extra  amount  of  injection  put  on. 

Fig.  26. 


DIAGRAM  FROM  ALLEN  S  ENGINE. 

The  pump  appears  to  be  quite  too  small  for  the  work  it  has  to 
do,  as  is  seen  hy  the  different  configuration  of  the  diagram  from 
that  of  the  diagrams  represented  in  figs.  11  and  18,  which  are 
also  diagrams  taken  from  air-pumps.  In  those  diagrams,  how- 
ever, the  stroke  of  the  bucket  is  more  than  half  performed,  be- 

Figs.  27  and  28. 


DIAGRAMS  FROM  ALLEN  S  ENGINE. 

fore  the  pressure  rises  above  the  atmospheric  line ;  whereas  in 
fig.  30,  the  pressure  rises  above  the  atmospheric  line  the  moment 
the  bucket  begins  to  ascend,  showing  that  at  that  time  the 
whole  of  the  pump  bai'rel  is  filled  with  water.  The  vacuum 
must  always  be  inferior  where  the  air-pump  is  gorged  with 
water. 


DIAGRAMS   TAKEN   FROM   AIR-PUMPS. 


359 


Unlike  the  previous  diagrams  taken  from  air-pumps,  we  see 
in  these  figures  the  pressure  or  resistance  has  to  be  encountered 
from  the  beginning,  or  nearly  the  beginning  of  the  stroke ;  and 
the  vacuum  is  not  good,  and  the  pump  overloaded.  There  is  a 


DIAGRAM  FROM  AIR-PUMP  OF  DUKE  OF  SUTHERLAND'S  YACHT. 
(OBDINABY  INJECTION.) 

worse  vacuum  with  the  increased  injection  than  with  the  ordi- 
nary injection,  showing  that  it  is  not  the  too  great  heat  of  the 
condenser  which  makes  the  vacuum  bad,  but  a  deficient  capacity 
of  pump,  or  an  imperfect  emptying  of  it  every  stroke. 

Fig.  30. 


DIAGRAM  FROM  AIR-PUMP  OF  DUKE  OF  SUTHERLAND  S  YACHT. 

(EXTRA  INJECTION  PUT  os.) 

In  fig.  31  we  have  a  diagram  illustrative  of  the  diminished 
load  upon  the  air-pump,  caused  by  putting  an  air-vessel  on  the 
hot-well.  A  is  the  atmospheric  line,  and  B  is  the  line  represent- 


360  POWER   AND   PERFORMANCE    OF   ENGINES. 

ing  the  ordinary  pressure  existing  in  the  hot-well  when  the  air- 
vessel  is  in  operation.  By  letting  out  the  air  the  pressure  rises 
to  c,  showing  that  the  pressure  on  the  pump  is  less  with  the  air- 
vessel  than  without  it.  If  the  air-vessel  be  discarded,  an  in- 
creased velocity  must  be  given  to  the  water  passing  through  the 
waste-water  pipe  to  enable  the  bucket  to  ascend,  and  this  im- 
plies a  waste  of  power. 

Fig.  si. 


DIAGRAM  FROM  HOT-WELL  OF  DUKE  OF  SUTHERLAND'S  YACHT. 
(AIR- VESSEL  ON.) 

In  fig.  32  we  have  a  diagram  taken  from  the  hot-well  of  the 
Duke  of  Sutherland's  yacht  after  the  air-vessel  has  been  re- 
moved. In  this  diagram  the  pressure  begins  to  rise  pretty 
quickly,  as  the  bucket  of  the  pump  ascends ;  and  the  maximum 
pressure,  when  reached,  is  maintained  pretty  uniform  to  the  end 
of  the  stroke.  It  does  not  then,  however,  suddenly  fall,  but 
only  gradually,  owing  to  the  momentum  of  the  water ;  and  the 

Fig.  32. 


DIAGRAM  TAKEN  FROM  HOT-WELL  OF  DUKE  OF  SUTHERLAND  S  TACHT. 
(AIB- VESSEL  OFF.) 

pencil  does  not  again  come  down  to  the  atmospheric  line  until 
nearly  half  the  downward  stroke  of  the  pump  has  been  com- 
pleted. 

In  fig.  33  we  have  a  diagram  taken  from  the  hot- well  of  the 
steamer  '  Scud,'  a  vessel  fitted  with  two  single-trunk  engines, 
that  is,  trunk  engines  with  the  trunks  projecting  only  at  one 
end,  and  not  at  both,  as  in  Messrs.  Penn's  arrangement.  The 


DIAGRAMS   TAKEN   FROM   WATER-PUMPS. 


361 


engines  are  angular,  working  up  to  the  screw-shaft,  and  the 
cylinders  are  68  inches  diameter,  and  4J-  feet  stroke.  The 
trunks  are  41  inches  diameter.  These  engines  made  42  strokes 
per  minute,  and  worked  up  to  8£  times  the  nominal  power. 
The  diagram  shows  an  increase  of  pressure  in  the  hot- well  at 

Fig.  33. 


DIAGRAM  TAKEN  FROM  HOT-WELL  OF  STEAMER  '  SCUD. 

each  end  of  the  stroke  of  the  double-acting  pump,  and  the 
pressure  runs  up  slowly  at  each  end  of  the  stroke,  when  it 
slowly  falls,  forming  the  loop  shown  in  the  diagram. 

Diagram  from  Pump  of  Water-works. — Fig.  34  is  a  diagram 
taken  from  the  pump  of  a  pumping-engine  at  the  Cork  Water- 
Fig.  34. 


V7V 


DIAGRAM  TAKEN  FROM  PUMP  OF  CORK  WATER-WORKS. 

works.  This  engine,  in  common  with  most  pumping-engines  of 
modern  construction,  is  a  rotative  engine — an  innovation  first 
effectually  introduced  by  Mr.  David  Thomson.  The  engines 
make  31  revolutions  per  minute,  and  work  with  steam  of  40  Ibs. 
on  the  square  inch.  "When  the  plunger  is  ascending,  the  pump 
15 


362      POWEK  AND  PERFORMANCE  OF  ENGINES 

is  sucking ;  and  when  the  piston  is  descending  it  is  forcing,  and 
the  diagram  shows  that  both  operations  are  accomplished  with 
much  regularity,  and  without  any  of  those  sudden  fluctuations 
which  always  occasion  a  loss  of  power. 

Having  now  shown  in  what  manner  the  indicator  may  be  ap- 
plied lo  ascertain  the  performance  of  ordinary  engines,  I  shall 
proceed  to  describe  the  manner  of  its  application  in  the  case  of 
double-cylinder  engines.  In  this  class  of  engines  the  steam 
having  pressed  the  first  piston  to  the  end  of  its  stroke  in  the 
manner  of  a  high-pressure  engine,  escapes,  not  into  the  atmos- 
phere, but  into  another  engine  of  larger  dimensions,  where  it 
expands,  and  acts  as  low-pressure  steam  on  the  piston  of  the 
second  engine,  being  finally  condensed  in  the  usual  manner. 
The  pressure  urging  the  first,  or  high-pressure  piston,  is  conse- 
quently the  difference  of  pressure  between  the  steam  in  the 
boiler  and  that  in  the  second  cylinder ;  and  the  pressure  urging 
the  second,  or  low-pressure  piston,  is  the  difference  of  pressure 
between  the  steam  on  the  eduction  side  of  the  high-pressure 
cylinder  and  that  of  the  vapour  in  the  condenser.  There  will  be 
a  small  difference  between  the  pressures  in  the  communicating 
parts  of  the  high  and  low-pressure  engines,  just  as  there  is  a 
small  difference  between  the  vacuum  in  the' cylinder  and  that 
in  the  condenser.  But  in  well-constructed  high-pressure  engines 
this  difference  will  not  sensibly  detract  from  the  power. 

Diagrams  from  Double-cylinder  Engines. — In  proceeding  to 
determine  the  power  of  a  double-cylinder  engine,  we  first  de- 
termine by  a  diagram  and  a  computation,  such  as  I  have  already 
given  examples  of,  the  power  exerted  by  the  high-pressure  en- 
gine ;  and  then,  in  like  manner,  we  determine  the  power  exerted 
by  the  low-pressure  engine.  The  total  power  is  obviously  the 
sum  of  the  two. 

An  example  of  the  diagrams  taken  from  the  high  and  low- 
pressure  cylinders  of  a  double-cylinder  engine,  at  the  Lambeth 
"Water-works,  constructed  by  Mr.  David  Thomson,  and  erected 
under  his  direction,  will  next  be  given.  In  a  paper  read  by  Mr. 
Thomson  before  the  Institution  of  Mechanical  Engineers,  and  a 
copy  of  which  he  has  forwarded  to  me,  the  main  particulars  of 


OF   THE    DOUBLE    CYLINDER   KIND.  363 

these  engines  are  recited ;  and  some  of  the  most  material  points 
of  that  paper  I  shall  here  recapitulate,  as  these  engines  consti- 
tute a  very  superior  example  of  the  double-cylinder  class  of 
engine. 

These  engines  are  heam-engines,  having  the  double  cylinders 
at  one  end  of  the  beam,  and  a  crank  and  connecting-rod  at  the 
other  end.  Four  engines  of  150  horse-power  each  are  fixed  side 
by  side  in  the  same  house,  arranged  in  two  pairs,  each  pair 
working  on  to  one  shaft,  with  cranks  at  right-angles,  and  a  fly- 
wheel between  them.  The  strokes  of  the  crank  and  of  the  large 
cylinder  are  equal ;  while  the  small  cylinder,  which  receives  the 
steam  direct  from  the  boiler,  has  a  shorter  stroke,  and  its  effec- 
tive capacity  is  nearly  one-fourth  that  of  the  large  cylinder.  The 
pumps  are  connected  direct  to  the  beams  near  the  connecting- 
rod  end  by  means  of  two  side  rods,  between  which  the  crank 
works.  The  pumps  are  of  the  combined  plunger  and  bucket 
construction,  and  are  thus  double-acting,  although  having  only 
two  valves.  This  kind  of  pump,  which  is  now  in  general  use, 
was  first  introduced  by  Mr.  Thomson  at  the  Eichmond  and  the 
Bristol  Water- works  in  the  year  1848.  The  following  are  the 
principal  dimensions  of  the  engines : — Diameter  of  large  cylin- 
der, 46  ins. ;  diameter  of  small  cylinder,  28  ins. ;  stroke  of  large 
cylinder,  8  ft. ;  stroke  of  small  cylinder,  5  ft.  6f  ins. ;  diameter 
of  pump-barrel  23$-  ins. ;  diameter  of  pump-plunger,  16$  ins. ; 
stroke  of  pump,  6  ft.  llf  ins. ;  length  of  beam  between  extreme 
centres,  26  ft.  6  ins. ;  height  of  beam-centre  from  floor,  21  ft. 
4  ins.  The  valves  are  piston- valves,  connected  by  a  hollow  pipe, 
through  which  the  escaping  steam  passes,  and  are  so  constructed 
that  one  valve  effects  the  distribution  of  the  steam  in  each  pair 
of  cylinders. 

The  cylinder-ports  are  rectangular,  with  inclined  bars  across 
the  faces  to  prevent  the  packing-rings  of  the  valve  from  catching 
against  the  edges  of  the  ports ;  and  the  bars  are  made  inclined 
instead  of  vertical,  in  order  to  avoid  any  tendency  to  grooving 
the  valve-packing.  The  openings  of  the  port  extend  two-thirds 
round  the  circumference  of  the  valve  in  the  ports  of  the  large 
cylinder ;  but  they  extend  only  half  round  in  the  ports  of  the 


364      POWER  AND  PERFORMANCE  OF  ENGINES. 

small  cylinder.  The  packing  of  the  valve  consists  of  the  four 
cast-iron  rings,  which  are  cut  at  one  side  exactly  as  in  an  ordi- 
nary piston,  the  joint  being  covered  by  a  plate  inside.  A  con- 
siderably stronger  pressure  of  the  rings  against  the  valve-chest 
is  required  than  was  at  first  expected,  because  the  openings  of 
the  steamports  extend  so  far  round  the  valve ;  and  for  this  pur- 
pose springs  are  placed  inside  the  packing-rings  to  assist  their 
own  elasticity.  This  construction  of  valve  has  the  advantage  of 
admitting  of  great  simplicity  in  the  castings  of  the  cylinders ; 
and  also  allows  of  the  whole  of  the  valve-work  being  executed 
in  the  lathe,  which  is  generally  the  cheapest  and  most  correct 
kind  of  work  in  an  engineering  workshop.  These  valves  are 
worked  by  cams. 

The  principal  object  aimed  at  in  the  construction  of  this 
piston-valve  was  a  reduction  to  a  minimum  of  the  loss  of  pres- 
sure which  the  steam  undergoes  in  passing  from  the  small  cyl- 
inder to  the  large  one.  This  is  here  accomplished  by  making 
the  passage  of  moderate  dimensions  and  as  direct  as  possible ; 
and  also  by  preventing  any  communication  of  this  passage  with 
the  condenser,  so  that  when  the  steam  from  the  small  cylinder 
enters  the  passage,  the  latter  is  already  filled  with  steam  of  the 
density  that  existed  in  the  large  cylinder  at  the  termination  of 
the  previous  stroke.  In  constructing  the  engines  some  doubt 
was  entertained  as  to  the  best  size  of  passage,  in  order  on  the 
one  hand  to  avoid  throttling  the  steam,  and  on  the  other  to  ob- 
viate as  much  as  possible  the  loss  of  steam  in  filling  the  passage. 
The  size  adopted  was  a  pipe  6  inches  in  diameter,  or  l-60th  of 
the  area  of  the  large  cylinder,  for  a  speed  of  piston  of  230  feet 
per  minute  in  the  large  cylinder :  and  this  is  believed  to  be 
about  the  best  proportion,  the  entire  cubic  content  of  the  whole 
passage  in  the  valve  amounting  to  3,944  cubic  inches.  The  indi- 
cator diagrams  show  that  with  this  construction  of  valve  there 
is  very  little  or  no  throttling  of  the  steam,  and  also  that  there  is 
but  a  very  moderate  drop  in  the  pressure  as  the  steam  passes 
from  the  small  cylinder  into  the  large  one.  In  this  respect  the 
valve  completely  answered  the  expectations  entertained  of  it, 
and  left  little  further  to  be  desired  on  this  point. 


DIAGRAMS   FROM   DOUBLE   CYLINDERS. 


365 


In  figs.  35  and  36  we  have  diagrams  taken  simultaneously 
from  the  top  of  the  small  cylinder  and  the  bottom  of  the  large 
one,  in  the  double-cylinder  engines  of  the  Lambeth  Water- 
works, designed  by  Mr.  Thomson— the  high-pressure  diagram 
being  placed  above,  and  the  low-pressure  diagram  below,  with  a 
small  space  between  the  two  answering  to  the  loss  of  pressure  in 
the  communicating  pipe.  The  dotted  line  shows  the  exhaust- 
line  in  the  small  cylinder  reversed,  so  as  to  tell  by  direct  measure- 
Figs.  35  and  36. 


DIAGRAMS  FROM  DOUBLE-CYLINDER  ENGINES,  LAMBETH  WATER-WORKS. 

(TAKEN  SIMU  I.TA  M:.  .rsi.v  FBOM  TOP  or  SMALL  •  v LIN :>!•  i:  AND  BOTTOM  OF  LABOB 

CYLINDER.) 

ment  between  this  bottom  and  the  top  of  the  diagram  what  is 
the  pressure  of  the  steam  on  the  small  piston  at  every  part  of 
its  stroke. 

The  most  material  of  the  results  which  may  be  deduced  from 
the  indicator  diagrams  of  this  engine  are  as  follows : — Percent- 
age of  stroke  at  which  steam  is  cut  off  in  small  cylinder,  40  per 
cent.;  total  expansion  at  end  of  stroke  in  small  cylinder,  in 
terms  of  bulk  before  expansion,  2 '41  per  cent. ;  amount  of  ex- 
pansion on  passing  from  small  to  large  cylinder,  in  terms  of  bulk 


3G6      POWER  AND  PERFORMANCE  OF  ENGINES. 

before  escaping  from  small  cylinder,  1'18  per  cent.;  total  expan- 
sion at  end  of  stroke  in  large  cylinder,  in  terms  of  original  bulk, 
9-66  per  cent. ;  total  amount  of  efficient  expansion,  in  terms  of 
original  bulk,  8'19  per  cent. ;  total  pressure  of  steam  per  square 
inch  at  point  of  cutting  off,  41  Ibs. ;  theoretical  total  pressure 
at  end  of  stroke  of  small  piston,  1TO  Ibs. ;  actual  total  pressure 
shown  by  diagram,  18'0  Ibs. ;  excess  of  actual  over  theoretical 
in  percentage  of  actual  pressure,  6  per  cent. ;  theoretical  loss  of 
pressure  in  passage  from  small  to  large  cylinder,  2'6  Ibs. ;  actual 
loss  shown  by  diagram,  4'5  Ibs. ;  theoretical  total  pressure  at 
end  of  stroke  of  large  piston,  4- 2  Ibs. ;  actual  total  pressure  shown 
by  diagram,  5*5  Ibs. ;  excess  of  actual  over  theoretical  in  per- 
centage of  actual  pressure,  23  per  cent.;  mean  pressure  on 
crank-pin  from  both  cylinders,  22,400  Ibs. ;  maximum  ditto,  36,- 
058  Ibs. ;  ratio  of  maximum  to  mean,  1'61  to  I'OO ;  ratio  of  max- 
imum to  mean  pressure  on  crank-pin  in  a  single  cylinder  engine 
with  the  same  total  amount  of  efficient  expansion,  the  clearances 
and  ports  bearing  the  same  proportion  to  the  working  capacity 
of  the  cylinder,  namely,  l-40th  part  (this  ratio  is  calculated  from 
the  ordinary  logarithmic  expansion  curve),  2*75  to  I'OO ;  effi- 
ciency of  steam  contained  in  large  cylinder  at  end  of  stroke,  as 
shown  by  diagram,  if  used  without  expansion,  taken  as  1*00 ; 
actual  efficiency  of  same  steam  as  employed  in  both  cylinders, 
as  shown  by  diagram,  2-90 ;  theoretical  efficiency  of  the  same 
steam  if  expanded  to  the  same  degree  as  the  total  amount  of 
efficient  expansion,  3'10.  The  engines  are  fitted  with  steam- 
jackets,  and  these  indicator  diagrams  show  that  the  pressure  of 
the  steam  at  the  end  of  the  stroke,  instead  of  falling  short  of 
what  it  ought  to  be  by  the  theoretical  expansion  curve,  exceeds 
that  amount  by  about  23  per  cent,  of  the  actual  final  pressure. 
It  might  be  supposed  that  the  increased  pressure  at  the  end  of 
the  stroke  was  due  to  the  heat  imparted  from  the  jackets  either 
superheating  the  steam  or  converting  the  watery  vapour  mixed 
with  it  into  true  steam ;  and  probably  the  latter  is  the  cause  of 
a  small  part  of  the  observed  effect ;  but  Mr.  Thomson  considers 
it  less  likely  that  sufficient  heat  could  be  communicated  from 
the  jackets  to  produce  an  increase  of  23  per  cent,  in  the  actual 


FEATURES    OF   WATER-WORKS    ENGINES.  367 

final  pressure,  especially  as  on  several  occasions  the  condensed 
water  from  the  jackets  has  been  collected  and  found  not  to  ex- 
ceed half-a-gaUon  per  hour.  The  experiments  made  on  the 
quantities  of  water  passed  from  the  boilers  give  uniformly  the 
result,  that  a  considerably  larger  quantity  of  water  passes  from 
the  boilers  than  is  accounted  for  by  the  indicator  diagrams, 
taking  the  quantity  and  pressure  of  the  steam  just  before  it 
escapes  to  the  condenser  as  the  basis  of  calculation.  In  some 
trials  made  within  a  few  days  of  these  diagrams  being  taken,  the 
excess  of  water  thus  disappearing  from  the  boilers  was  about 
37  per  cent.  To  suppose  that  the  valve  was  leaking  might  ac- 
count for  it  ;*  but  besides  great  care  having  been  taken  to  avoid 
this  source  of  error,  it  can  hardly  be  supposed  that  the  valve 
was  always  leaking  more  than  the  pistons. 

To  ascertain  the  amount  of  friction  in  these  engines  Mr. 
Thomson  made  many  experiments,  and  found  that,  when  the 
engines  were  new,  and  working  at  perhaps  little  more  than  half 
their  power,  the  loss  in  comparing  the  work  done  with  the  indi- 
cator diagrams  amounts  to  as  much  as  25  per  cent,  of  the  indi- 
cated power ;  but  in  these  cases  the  pistons  have  been  too  tight 
in  the  cylinders,  and  when  this  error  has  been  corrected,  and  the 
engines  worked  up  to  their  regular  work,  all  the  losses  were 
brought  down  to  from  12  to  15  per  cent,  of  the  indicated  power. 
This  includes  the  friction  of  both  the  engines  and  the  pumps, 
the  working  of  the  air-pumps,  feed-pumps,  cold-water  pumps, 
and  pumps  for  charging  the  air-vessels  with  air. 

With  regard  to  the  economy  of  fuel  attained  by  these  double- 
cylinder  engines,  it  may  be  stated  that  the  four  pumping-engines 
at  the  Lambeth  "Water- works  are  fixed  in  one  house,  and  are 
employed  in  pumping  through  a  main-pipe  30  inches  diameter 
and  about  nine  miles  in  length ;  and  when  all  the  engines  are 
working  together  at  their  ordinary  speed  of  14  revolutions  per 
minute,  the  lift  on  the  pumps,  as  measured  by  a  mercurial  gauge, 
is  equal  to  a  head  of  about  210  feet  of  water.  Under  these  cir- 
cumstances they  were  tested  by  Mr.  Field  soon  after  being  fin- 

*  Some  of  the  disappearance  of  the  heat  is  no  doubt  impntable  to  its  transfor- 
mation into^ower,  as  explained  under  the  head  of  thermo-dynamics. 


3G8      POWER  AND  PERFORMANCE  OF  ENGINES 

ished,  in  a  trial  of  24  hours'  duration  without  stopping.  The 
actual  work  done  by  the  pumps  during  this  trial  was  equal  to 
97,064,894  Ibs.,  raised  one  foot  high  for  every  112  Ibs.  of  coal 
consumed ;  in  addition  to  which  this  consumption  included  the 
friction  of  the  engines  and  pumps,  and  the  power  required  to 
work  the  air-pumps,  feed  and  charging-pumps,  and  the  pumps 
raising  the  water  for  condensation.  The  coal  used  was  "Welsh, 
of  good  average  quality. 

The  economy  in  consumption  of  fuel  during  this  trial,  and  in 
the  subsequent  regular  working  of  these  engines,  together  with 
the  satisfactory  performance  generally  of  the  engines  and  pump 
work,  induced  the  Chelsea  "Water- works  Company,  and  also  the 
New  Eiver  Company,  each  to  erect  in  1854  a  set  of  four  similar 
engines,  which  were  made  almost  exactly  the  same  as  the  Lam- 
beth Water- works  engines  already  described,  with  the  exception 
that  a  jacket  of  high-pressure  steam  was  in  these  subsequent  en- 
gines provided  under  the  bottoms  of  the  cylinders,  which  had 
not  been  done  with  the  previous  engines.  The  pumps  were 
also  different  in  size  to  suit  the  different  lifts. 

The  New  Eiver  engines  were  tested  soon  after  being  com- 
pleted, and  the  result  reported  was  113  million  Ibs.  raised  one 
foot  high  by  112  Ibs.  of  "Welsh  coal.  But  this  duty  was  obtained 
from  a  trial  of  only  seven  or  eight  hours'  duration,  which  is  too 
short  to  obtain  very  trustworthy  results. 

The  set  of  engines  made  for  the  Chelsea  "Water-works  was  the 
last  finished,  and  on  completion  the  engines  were  tested  by  Mr. 
Field  in  the  same  manner  as  the  Lambeth  engines,  by  a  trial  of 
24  hours'  continuous  pumping.  The  coal  used  was  Welsh,  as  be- 
fore, and  the  duty  reported  was  103'9  million  Ibs.  raised  one  foot 
high  by  112  Ibs.  of  coal.  This,  as  in  the  previous  instance,  was 
the  duty  got  from  the  pumps  in  actual  work  done,  no  allowance 
being  made  for  the  friction  of  the  engines  and  pumps,  and  the 
power  required  to  work  the  air-pumps,  cold-water  pumps,  &c. 
At  the  time  of  these  engines  being  tested,  the  loss  by  friction 
and  by  working  the  air-pumps,  &c.,  averaged  about  20  per  cent. 
of  the  power,  as  given  by  the  indicator  diagrams ;  so  that  if  the 
duty  had  been  estimated  from  the  indicator  diagrams,  fts  is  usual 


OF    THE    DOUBLE    CYLINDER   KIND.  369 

in  marine  engines,  it  would  have  been  103'9  x  J^,  or  about  130 
million  Ibs.  raised  one  foot  by  112  Ibs.  of  coal,  which  is  equiva- 
lent to  a  consumption  of  1-T  lb.  per  indicated  horse-power  per 
hour. 

In  figs.  37  and  38  we  have  diagrams  taken  from  a  small  en- 
gine called  Wenham's  double-cylinder  engine,  working  with  a 
pressure  of  40  Ibs.  per  square  inch  in  the  boiler,  and  exhibited  at 
the  Great  Exhibition  in  1862.  The  average  pressure  on  the 

Pig.  3T. 


DIAGRAM  FROM   HIGH-PRESSURE     CYLINDER  OF  WENHAM's  DOUBLE-CYLINDER 
ENGINE. 

(CYLINDER  THREE  INCHES  DIAMETER  AND  TWELVE  INCHES  STROKE.) 

piston  of  the  high-pressure  engine,  which  is  3  inches  diameter 
and  12  inches  stroke,  is  26'61bs.  per  square  inch,  and  the  power 
it  exerts  is  3*16  horses.  The  average  pressure  exerted  on  the 

Fig.  38. 


DIAGRAM   FROM   LOW-PRESSURE   CYLINDER  OF   WENHAM's    DOUBLE-CYLINDER 
ENGINE. 

piston  of  the  low-pressure  engine  is  8'5  Ibs.  per  square  inch,  and 
the  power  it  exerts  is  2*37  horses.  The  steam  in  passing  from 
one  cylinder  to  the  other  is  heated  anew,  as  had  previously  been 
done  by  me  in  the  engines  of  the  steamer  'Jumna,'  of  400  horse- 
power. The  total  power  developed  in  both  cylinders  of  Wen- 
ham's  engine  is  6*05  horses. 

Having  now  explained  how  to  interpret  a  diagram,  the  next 
thing  is  to  explain  how  to  take  one,  and  here  I  cannot  do  better 
15* 


370      POWER  AND  PERFORMANCE  OF  ENGINES. 

than  recite   the  instructions  for  this    operation  issued    with 
Eichards'  indicator  by  the  makers,  Elliot  Brothers,  of  the  Strand. 

To  fix,  the  Paper. — Take  the  outer  cylinder  off  from  the  instrument, 
secure  the  lower  edge  of  the  paper,  near  the  corner,  by  one  spring,  then 
bend  the  paper  round  the  cylinder,  and  insert  the  other  corner  between 
the  springs.  The  paper  should  be  long  enough  to  let  each  end  project 
at  least  half-an-inch  between  the  springs.  Take  the  two  projecting  ends 
with  the  thumb  and  finger,  and  draw  the  paper  down,  taking  care  that 
it  lies  quite  smooth  and  tight,  and  that  the  corners  come  fairly  together, 
and  replace  the  cylinder.  The  spring  used  on  this  indicator  for  holding 
the  paper  will  be  found  preferable  to  the  hinged  clamp.  A  little  prac- 
tice, with  attention  to  the  above  directions,  will  enable  any  one  to  fix 
the  paper  very  readily. 

Tlit,  marking-point  should  be  fine  and  smooth,  so  as  to  draw  a  fine 
line,  but  not  cut  the  paper.  It  may  be  made  of  a  brass  wire  ;  the  best 
material  is  gun-metal,  which  keeps  sharp  for  a  long  time,  and  the  line 

Fig.  39. 


made  by  it  is  very  durable.  Lines  drawn  by  German  silver  points  are 
liable  to  fade.  A  large-sized  common  pin,  a  little  blunted,  answers  for 
a  marking-point  very  well  indeed ;  a  small  file  and  a  bit  of  emery  cloth 
used  occasionally  will  keep  the  point  in  order. 

To  connect  the  Cord,. — The  indicator  having  been  attached,  and  the 
correct  motion  obtained  for  the  drum,  and  the  paper  fixed,  the  next 
thing  is  to  see  that  the  cord  is  of  the  proper  length  to  bring  the  diagram 
in  its  right  place  on  the  paper — that  is,  midway  between  the  springs 
which  hold  the  paper  on  the  drum.  In  order  to  connect  and  disconnect 
readily,  the  short  cord  on  the  indicator  is  furnished  with  a  hook,  and  at 
the  end  of  the  cord  coming  from  the  engine  a  running  loop  maybe  rove 
in  a  thin  strip  of  metal,  in  the  manner  shown  in  the  preceding  cut,  by 
which  it  can  be  readily  adjusted  to  the  proper  length,  and  taken  up  from 
time  to  time,  as  it  may  become  stretched  by  use.  On  high-speed  en- 
gines, it  is  as  well,  instead  of  using  this,  to  adjust  the  cord  and  take  up 
the  stretching,  as  it  takes  place,  by  tying  knots  in  the  cord.  If  the  cord 
becomes  wet  and  shrinks,  the  knots  may  need  to  be  untied,  but  this 


METHOD   OP  TAKING   A   DIAGRAM.  371 

rarely  happens.  The  length  of  the  diagram  drawn  at  high  speeds  should 
not  exceed  four  and  a-half  inches,  to  allow  changes  in  the  length  of  the 
cord  to  take  place  to  some  extent,  without  causing  the  drum  to  revolve 
to  the  limit  of  its  motion  in  either  direction.  On  the  other  hand,  the 
diagram  should  never  be  drawn  shorter  than  is  necessary  for  this 
purpose. 

To  take  the  Diagram. — Every  thing  being  in  readiness,  turn  the  han- 
dle of  the  stop-cock  to  a  vertical  position,  and  let  the  piston  of  the  in- 
dicator play  for  a  few  moments,  while  the  instrument  becomes  warmed. 
Then  turn  the  handle  horizontally  to  the  position  in  which  the  commu- 
nication is  opened  between  the  under  side  of  the  piston  and  the  atmos- 
phere, hook  on  the  cord,  and  draw  the  atmospheric  line.  Then  turn 
the  handle  back  to  its  vertical  position,  and  take  the  diagram.  When 
the  handle  stands  vertical,  the  communication  with  the  cylinder  is  wide 
open,  and  care  should  be  observed  that  it  does  stand  in  that  position 
whenever  a  diagram  is  taken,  so  that  this  communication  shall  not  be 
in  the  least  obstructed. 

To  apply  the  pencil  to  the  paper,  take  the  end  of  the  longer  brass  arm 
with  the  thumb  and  forefinger  of  the  left  hand,  and  touch  the  point  as 
gently  as  possible,  holding  it  during  one  revolution  of  the  engine,  or 
during  several  revolutions,  if  desired.  There  is  no  spring  to  press  the 
point  to  the  paper,  except  for  oscillating  cylinders ;  the  operator,  after 
admitting  the  steam,  waits  as  long  as  he  pleases  before  taking  the  dia- 
gram, and  touches  the  pencil  to  the  paper  as  lightly  as  he  chooses.  Any 
one,  by  taking  a  little  pains,  will  become  enabled  to  perform  this  opera- 
tion with  much  delicacy.  As  the  hand  of  the  operator  cannot  follow  the 
motions  of  an  oscillating  cylinder,  it  is  necessary  that  the  point  be  held 
to  the  paper  by  a  light  spring,  and  instruments  to  be  used  on  engines 
of  this  class  are  furnished  with  one  accordingly. 

Diagrams  should  not  be  taken  from  an  engine  until  some  time  after 
starting,  so  that  the  water  condensed  in  warming  the  cylinder,  Ac., 
shall  have  passed  away.  Water  in  the  cylinder  in  excess  always  distorts 
the  diagram,  and  sometimes  into  very  singular  forms.  The  drip-cocks 
should  be  shut  when  diagrams  are  being  taken,  unless  the  boiler  is 
priming.  If  when  a  new  instrument  is  first  applied  the  line  should 
show  a  little  evidence  of  friction,  let  the  piston  continue  in  action  for  a 
short  time,  and  this  will  disappear. 

As  soon  as  the  diagram  is  taken,  unhook  the  cord ;  the  paper  cylin- 
der should  not  be  kept  in  motion  unnecessarily,  as  it  only  wears  out  the 
spring,  especially  at  high  velocities.  Then  remove  the  paper,  and 
minute  on  the  back  of  it  at  once  as  many  of  the  following  particulars  as 
you  have  the  means  of  ascertaining,  viz. : 

The  date  of  taking  the  diagram,  and  scale  of  the  indicator. 


372  POWER   AND   PERFORMANCE   OF   ENGINES. 

The  engine  from  which  the  diagram  is  taken,  which  end,  and  which 
engine,  if  one  of  a  pair. 

The  length  of  the  stroke,  the  diameter  of  the  cylinder,  and  the  num- 
ber of  double  strokes  per  minute. 

The  size  of  the  ports,  the  kind  of  valve  employed,  the  lap  and  lead 
of  the  valve,  and  the  exhaust  lead. 

The  amount  which  the  waste-room,  in  clearance  and  thoroughfares, 
adds  to  the  length  of  the  cylinder. 

The  pressure  of  steam  in  the  boiler,  the  diameter  and  length  of  the 
pipe,  the  size  and  position  of  the  throttle  (if  any),  and  the  point  of  cut- 
off. 

On  a  locomotive,  the  diameter  of  the  driving-wheels,  and  the  size  of 
the  blast  orifice,  the  weight  of  the  train,  and  the  gradient,  or  curve. 

On  a  condensing-engine,  the  vacuum  by  the  gauge,  the  kind  of  con- 
denser employed,  the  quantity  of  water  used  for  one  stroke  of  the  en- 
gine, its  temperature,  and  that  of  the  discharge,  the  size  of  the  air-pump 
and  length  of  its  stroke,  whether  single  or  double  acting,  and,  if  driven 
independently  of  the  engine,  the  number  of  its  strokes  per  minute,  and 
the  height  of  the  barometer. 

The  description  of  boiler  used,  the  temperature  of  the  feed-water,  the 
consumption  of  fuel  and  of  water  per  hour,  and  whether  the  boilers, 
pipes,  and  engine  are  protected  from  loss  of  heat  by  radiation,  and  if 
so,  to  what  extent. 

In  addition  to  these,  there  are  often  special  circumstances  which 
should  be  noted. 

Counter  and  Dynamometer. — There  are  other  instruments 
besides  the  indicator  for  telling  the  performance  of  an  engine — 
the  counter  which  registers  the  number  of  strokes  made  by  an 
engine  being  nsed  for  this  purpose,  in  the  case  of  pumping- 
engines,  working  with  a  uniform  load,  and  the  dynamometer 
being  employed  in  testing  the  power  exerted  by  small  engines. 
The  dynamometer  consists  of  a  moving  disc  well  oiled,  and  en- 
circled by  a  stationary  hoop,  which  can  be  so  far  tightened  as  to 
create  sufficient  friction  to  constitute  the  proper  load  for  the  en- 
gine. The  hoop  is  prevented  from  revolving  with  the  disc  by  an 
arm  extending  from  it,  which  is  connected  with  a  spring,  the 
tension  on  which,  reduced  to  the  diameter  of  the  disc,  represents 
the  load  which  the  friction  creates ;  and  the  load  multiplied  by 
the  space  passed  through  per  minute  by  any  point  on  the  cir- 
cumference of  the  disc  will  represent  the  power.  Such  dyna- 


NEW    FORM    OF   DUTY   METER. 


373 


mometers,  however,  cannot  be  conveniently  applied  to  large  en- 
gines ;  and  as  in  steam-vessels,  where  economy  of  fuel  is  most 
important,  the  counter  will  not  accurately  register  the  work 
done,  seeing  that  the  resistance  is  not  uniform,  and  as  without 
some  reliable  means  of  determining  the  power  produced  in  dif- 
ferent vessels  relatively  with  the  fuel  consumed,  it  is  impossible 
to  establish  such  a  comparison  of  efficiency  as  will  lead  to  emula- 
tion, and  consequent  improvement,  I  have  felt  it  necessary  to 
contrive  a  species  of  continuous  indicator,  or  power-metor,  for 

Fig.  40. 


BOUENK'S  DUTT  METER. 

ascertaining  and  recording  the  amount  of  work  done  by  any 
engine  during  a  given  period  of  time.  The  outline  of  one  form 
of  this  instrument  is  exhibited  in  fig.  40 ;  but  I  prefer  that  the 
cylinder  should  be  horizontal  instead  of  vertical,  and  that  it 
should  be  larger  in  diameter,  and  shorter — this  figure  being 
copied  from  a  photograph  of  an  instrument  I  had  converted  from 
a  common  M'Naught's  indicator,  for  the  sake  of  readiness  of 
construction.  In  this  instrument  one  end  of  the  indicator  cylin- 
der communicates  with  one  end  of  the  main  cylinder,  and  the 
other  end  of  the  indicator  cylinder  with  the  other  end  of  the 


374  POWER   AND   PERFORMANCE    OF   ENGINES. 

main  cylinder,  so  that  the  atmosphere  does  not  press  upon  the 
piston  of  the  indicator  at  all,  but  that  piston  is  pressed  on  either 
side  by  steam  or  vapour  of  precisely  the  same  tension  as  that 
which  presses  on  either  side  of  the  piston  of  the  engine.  The 
indicator  piston  is  pressed  alternately  upward  and  downward 
against  a  spring  in  the  usual  manner.  A  double-ended  lever  vi- 
brating on  a  central  pivot,  and  with  a  slot  carried  along  it  near- 
ly from  end  to  end,  as  in  the  link  of  a  common  link-motion,  is 
attached  to  the  side  of  the  cylinder,  and  from  this  slot  a  horizon- 
tal rod  extends  to  the  arm  of  a  ring  encircling  a  ratchet-wheel, 
there  being  a  number  of  pawls  in  this  ring  of  different  lengths  to 
engage  the  ratchets.  This  link  is  moved  backwards  and  forwards 
on  its  centre,  8  or  10  times  every  stroke  of  the  engine,  by  means 
of  the  lower  horizontal  rod  which  is  attached  at  one  end  to  the 
lower  end  of  the  link,  and  at  the  other  end  to  a  small  pin  in  the 
side  of  a  drum,  which  is  drawn  out  by  a  string,  like  the  drum 
for  carrying  the  paper  in  a  common  indicator,  and  is,  in  like 
manner,  returned  by  a  spring ;  but  the  dimensions  of  the  drum, 
and  the  place  of  attachment  of  the  string,  are  such  that  the  drum 
makes  a  considerable  number  of  turns — say  10 — for  each  stroke 
of  the  engine,  and  the  link  makes  the  same  number  of  recipro- 
cations. If  there  be  an  equality  of  pressure  on  each  side  of  the 
piston,  the  end  of  the  rod  moving  in  the  slot  will  be  in  the  mid 
position  ;  and  as  while  it  is  there  no  amount  of  vibration  of  the 
link  will  give  it  any  end  motion,  there  will  be  no  motion  under 
such  circumstances  communicated  to  the  ratchet.  If,  however, 
the  pressure  either  upward  or  downward  is  considerable,  the 
end  of  the  rod  will  be  moved  so  much  up  or  down  in  the  link 
that  its  reciprocation  will  give  considerable  end  motion  to  the 
rod  communicating  with  the  ratchet ;  and  the  amount  of  motion 
given  to  the  ratchet  every  stroke  will  represent  the  amount  of 
mean  pressure  urging  the  piston.  The  number  of  revolutions  to 
be  made  by  the  drum  every  stroke  having  been  once  definitively 
fixed,  it  is  clear  that  the  number  of  revolutions  it  will  make  per 
minute  will  depend  on  the  number  of  strokes  made  per  minute 
by  the  engine,  and  the  revolutions  of  the  ratchet-wheel  will 
consequently  represent  both  the  mean  pressure  and  the  speed  of 


HEATING    SURFACE    IN   MODERN    BOILERS.  375 

piston — or  in  other  words,  it  will  represent  the  power.  The 
spindle  of  the  ratchet  wheel  is  formed  into  a  screw,  which  works 
into  the  periphery  of  a  wheel  that  gives  motion  to  other  wheels 
and  hands,  like  the  train  of  a  gas-meter  ;  and  on  opening  the  in- 
strument at  the  end  of  any  given  time,  such  as  at  the  termina- 
tion of  a  voyage  of  an  ocean  steamer,  the  power  which  the  ves- 
sel has  exerted  since  she  started  on  the  voyage  will  be  found  to 
be  accurately  registered.  This  being  compared  with  the  quan- 
tity of  coals  consumed,  which  can  easily  be  found  from  the  books 
of  the  owners,  will  give  the  duty  of  the  engine ;  and  by  ascer- 
taining and  publishing  the  duty  of  different  vessels,  a  wholesome 
emulation  would  be  excited  among  engine-makers  and  engine 
tenders,  and  a  vast  reduction  in  the  consumption  of  fuel  would 
no  doubt  be  obtained.  For  many  years  past  I  have  urged  the 
introduction  of  that  system  of  registration  in  the  case  of  steam- 
vessels  which  in  the  case  of  the  Cornish  engines  speedily  led  to 
such  unprecedented  economy.  But  the  want  of  a  suitable  register- 
ing apparatus  constituted  a  serious  impediment,  and  I  have  con- 
sequently undertaken  to  contrive  the  instrument  of  which  a 
rough  outline  is  given  above. 

Heating  Surface  in  modern  Boilers. — The  quantity  of  heat- 
ing surface  given  in  modern  boilers  per  nominal  horse-power  has 
been  constantly  increasing,  until,  in  some  of  the  boilers  of  recent 
steam- vessels  intended  to  maintain  a  high  rate  of  speed,  it  has 
become  as  much  as  35  square  feet  per  nominal  horse-power ;  and 
such  vessels  exert  a  power  nine  times  greater  than  the  nominal 
power.  The  nominal  power,  in  fact,  has  ceased  to  be  any  measure 
of  the  dimensions  of  a  boiler ;  and  the  best  course  will  be  to  con- 
sider only  the  water  evaporated.  In  modern  marine  boilers  it 
may  be  reckoned  that  a  cubic  foot  of  water  will  be  evaporated 
in  the  hour  by  Y  Ibs.  of  coal  burned  on  70  square  inches  of  fire- 
bars, and  the  heat  from  which  is  absorbed  by  10  square  feet  of 
heating  surface,  so  that  the  consumption  of  coal  per  hour,  on  each 
square  foot  of  grate,  will  bo  14'4  Ibs.  If  the  steam  be  cut  off 
from  the  cylinder  when  one-third  of  the  stroke  has  been  per- 
formed, as  is  a  common  practice,  the  efficiency  of  the  steam 
will  be  somewhat  more  than  doubled,  or  a  horse-power  will  be 


376  POWER   AND    PERFORMANCE    OF    ENGINES. 

generated  with  something  less  than  3J  Ibs  of  coal.  In  large 
boilers  and  engines,  however,  the  efficiency  is  greater  than  in 
small,  and  there  is  further  benefit  obtained  from  superheating, 
and  from  heating  the  feed-water  very  hot.  In  modern  steam- 
vessels  of  efficient  construction,  therefore,  the  consumption  of  coal 
is  not  more  than  2^  Ibs.  per  actual  horse-power.  Boulton  and 
"Watt  put  sufficient  lap  upon  their  valves  to  cut  off  the  steam 
when  two-thirds  of  the  stroke  have  been  performed  as  a  minimum 
of  expansion ;  and  then,  by  aid  of  the  link-motion,  they  can  ex- 
pand still  more,  if  required,  so  as  to  cut  off  when  one-third  of 
the  stroke  has  been  performed. 

The  area  of  the  back  uptake  should  be  1 5  square  inches  per 
cubic  foot  evaporated ;  the  area  of  the  front  uptake  12  square 
inches,  and  the  area  of  the  chimney  7  square  inches  per  cubic  foot 
evaporated.  These  proportions  will  enable  the  dimensions  of  any 
boiler  to  be  determined  when  the  rate  of  expansion  has  been  fixed. 

The  proportion  in  which  the  actual  exceeds  the  nominal 
power  varies  very  much  in  different  engines,  but  about  4  or  4£ 
times  appears  to  be  the  prevalent  proportion  in  1865,  though,  as  I 
have  stated,  in  special  cases  twice  this  proportion  of  power  is 
exerted,  and  the  boilers  are  proportioned  to  give  the  increased 
supply  of  steam  required.  For  any  temporary  purpose  the  power 
may  be  increased  by  quickening  the  draught  through  the  furnace 
by  a  jet  of  steam  in  the  chimney;  but  in  such  case  the  consump- 
tion of  fuel  per  cubic  foot  of  water  evaporated  will  be  somewhat  in- 
creased. The  first  proportion  of  heating  surface,  however,  which 
the  flame  encounters  is  very  much  more  efficient  than  the  last 
portion,  in  consequence  of  the  higher  temperature  to  which  it  is 
subjected ;  and  if  the  draught  be  quickened  the  temperature  will 
be  increased,  and  every  square  foot  of  heating  surface  will  thereby 
acquire  a  greater  absorbing  power.  The  hotter  the  furnace  is, 
the  more  heat  will  be  absorbed  by  the  water  in  the  region  of  the 
furnace;  and  the  more  heat  that  is  absorbed  by  the  furnace  the 
less  will  be  left  for  the  tubes  to  absorb.  It  is  material,  therefore, 
to  maintain  high  bridges,  a  rapid  draught,  and  all  other  aids  to  a 
high  temperature  in  the  furnace ;  as  the  absorption  of  heat  will 
thus  be  more  rapid,  and  the  combustion  will  be  more  perfect, 


ADVANTAGES    OF   HOT    FUBNACES.  377 

from  the  high  temperature  to  which  the  smoke  is  exposed.  It 
will  increase  the  efficacy  of  the  heating  surface,  moreover,  if  the 
smoke  he  made  to  strike  against  instead  of  sliding  over  it ;  and 
this  end  will  he  best  attained  by  using  vertical  tubes,  with  the 
water  within  them,  on  which  the  smoke  may  strike  on  its  way 
to  the  chimney.  Such  tubes,  furthermore,  are  eligible  in  con- 
sequence of  the  facilities  they  give  for  the  rapid  circulation  of 
the  water  within  the  boiler ;  and  this  rapid  circulation  will  not 
merely  render  the  boiler  more  durable  by  preventing  overheat- 
ing of  the  metal,  but  as  the  rapidly  ascending  current,  by  carry- 
ing off  the  steam  and  presenting  a  new  surface  of  water  to  be  acted 
•upon,  keeps  the  metal  of  the  tubes  cool,  they  are  in  a  better  con- 
dition for  absorbing  heat  from  the  smoke  than  if  the  metal  had 
become  overheated  from  the  entanglement  of  steam  in  contact 
with  it,  which  impeded  the  access  of  the  water,  and  prevented 
the  rapid  absorption  of  heat  which  would  otherwise  take  place. 
In  locomotive  boilers,  where  the  temperature  of  the  furnace  is 
very  high,  as  much  evaporative  efficacy  is  obtained  from  7  Ihs.  of 
coal,  with  5  or  6  square  feet  of  heating  surface,  as  is  obtained  in 
land  and  marine  boilers  with  9  or  10 ;  and  the  reason  manifestly 
is,  that  as  the  rapidity  of  the  transmission  of  heat  increases  as  the 
square  of  the  temperature,  a  square  foot  of  heating  surface  in  a  fur- 
nace twice  as  hot  will  be  four  times  more  effective,  so  that  the 
tubes  are  left  with  comparatively  little  work  to  do,  from  so  much  of 
the  work  having  been  done  in  the  furnace.  Each  square  foot  of 
tube  surface  in  locomotives  will  only  evaporate  as  much  as  each 
square  foot  in  an  ordinary  land  and  marine  boiler ;  but  the  mean 
efficacy  of  the  whole  heating  surface  is,  nevertheless,  raised  very 
high  by  the  greatly  increased  efficacy  of  the  fire-box  surface, 
from  its  high  temperature.  It  is  desirable  to  imitate  these  con- 
ditions in  marine  and  land  furnaces  by  making  the  area  fire-grate 
small,  the  draught  rapid,  and  the  bridges  high,  to  the  end  that  a 
high  temperature  in  the  furnace  may  be  preserved,  and  a  con- 
sequently rapid  generation  of  steam  promoted.  It  would  also  be 
desirable,  and  not  difficult,  to  feed  the  furnaces  with  hot  air  instead 
of  with  cold,  which  would  conduce  more  to  economy  than  feeding 
the  boiler  with  hot  instead  of  cold  water ;  and  it  would  not  be  dif- 


378  POWER  AND   PERFORMANCE   OF   ENGINES. 

ficult  to  carry  out  this  improvement,  by  encircling  the  chimney 
with  air-casing  nearly  to  the  top,  and  conducting  the  air  which 
would  be  admitted  by  openings  around  the  casings  at  its  upper  end, 
past  the  smoke-box  doors,  to  the  end  of  the  furnaces.  The  only  diffi- 
culty which  might  be  apprehended  from  this  procedure  would  be 
the  increased  heat  and  diminished  durability  of  the  furnace-bars. 
But  this  difficulty  might  no  doubt  be  surmounted  by  making  the 
bars  deep  and  thin,  and  by  not  increasing  the  temperature  of  the 
entering  air  beyond  the  point  which  experience  proved  it  could  be 
raised  to  with  impunity.  The  area  of  the  casing  around  the  chim- 
ney would  require  to  be  about  as  great,  at  the  largest  part,  as  the 
area  of  the  chimney  itself.  But  it  could  be  made  conical,  or 
tapering  off  at  the  top,  and  the  air  might  be  admitted  in  vertical 
slits  extending  downwards  for  a  certain  length,  as  the  heat  at  the 
top  of  the  chimney  could  be  abstracted  by  such  a  small  volume 
of  air  as  a  narrow  casing  would  contain.  In  this  heating  of  the  air 
entering  furnaces  there  is  an  expedient  of  economy  available  for 
the  engineer  which  has  not  yet  been  brought  into  force ;  and  its 
effect  will  be  both  to  reduce  the  consumption  of  the  fuel  and  to 
render  the  existing  heating  surface  more  effective.  If,  for  ex- 
ample, we  take  the  existing  temperature  of  the  furnace  to  be 
3,000°  Fahrenheit,  and  if  we  increase  the  temperature  of  the  en- 
tering air  by  500°,  which  we  might  easily  do  without  any  new 
expense,  we  shall  not  merely  save  one-sixth  of  the  fuel,  but  we 
shall  render  the  absorbing  surface  of  the  furnace  more  efficacious 
by  raising  the  temperature  from  3,000°  to  3,500°.  Nor  will 
this  probably  be  the  limit  of  benefit  obtained ;  and  as  in  feeding 
boilers  with  boiling  water  instead  of  cold,  and  in  surrounding 
cylinders  by  steam  to  keep  them  hot  instead  of  exposing  them 
to  the  atmosphere,  we  obtain  a  greater  benefit  than  theory  would 
have  led  us  to  expect,  so  in  feeding  furnaces  with  hot  air  instead 
of  cold  air  we  shall  in  all  probability  obtain  a  larger  benefit  than 
that  which  theory  indicates.  The  experience  already  obtained 
of  the  saving  effected  by  using  the  hot  blast  in  iron  smelting 
furnaces  certainly  points  to  the  probability  of  such  a  realization ; 
and  one  manifest  effect  will  be,  that  the  combustion  of  the  coal 
will  be  rendered  more  perfect,  and  less  smoke  will  be  produced. 


DESIDERATA   AT   THE    PRESENT    TIME.  379 

The  present  system  of  land  and  marine  boilers,  however,  is 
altogether  faulty,  and  must  be  changed  completely.  When  I 
planned  and  constructed  the  first  marine  tubular  boiler  in  1838, 
and  which  was  adapted  for  working  with  a  high  pressure  of 
steam,  and  which  also  had  the  advantage  of  surface  condensation, 
the  innovation  was  a  step  in  advance,  and  it  has  proved  successful 
and  serviceable,  though  up  to  the  present  time  the  system  then 
propounded  by  me  has  not  been  fully  wrought  out  in  practice. 
But  we  now  want  something  much  better  than  what  would  have 
sufficed  for  our  wants  in  1838,  and  I  will  here  briefly  recapitulate 
what  we  require  and  must  obtain.  First,  then,  we  must  have  a 
still  higher  pressure  of  steam  than  I  contemplated  in  1838 ;  to 
obtain  which  with  safety  we  must  have  two  things ;  a  very  strong 
boiler,  and  absolute  immunity  from  salting.  The  expedient  of 
surface  condensation,  which  I  propounded  in  1838,  as  the  means 
of  accomplishing  the  last  disideratuni,  though  effectual  for  the 
purpose,  and  now  widely  adopted,  is  less  eligible  for  moderate 
pressures  than  the  method  of  preventing  salting  which  I  have 
since  suggested,  and  which  consists  in  the  introduction  of  a 
small  jet  in  the  eduction-pipe,  the  water  of  which,  though  unable 
wholly  to  condense  the  steam,  will  be  itself  raised  to  the  boiling 
point,  and  be  transmitted  to  the  boiler  without  any  means  of 
stopping  it  off;  and  the  excess  of  feed- water  which,  under  this 
arrangement,  will  always  be  entering  the  boiler,  will  escape 
through  a  continuous  blow  off,  and  thus  prevent  the  boiler  from 
salting.  The  column  of  steam  escaping  to  the  condenser  will, 
under  suitable  arrangements,  itself  force  this  water  into  the 
boiler ;  and  in  locomotives,  in  like  manner,  the  water  may  be 
forced  into  the  boiler  by  using  a  portion  of  the  steam  escaping 
from  the  blast  pipe  for  that  purpose,  whereby  the  boiler  will  be 
fed  with  boiling  water  by  the  aid  of  steam  otherwise  going  to 
waste.  In  this  way  marine  boilers  may  be  kept  from  salting ;  for 
the  sulphate  of  lime  which  is  deposited  from  sea  water  at  the  tem- 
peratures of  high-pressure  steam,  may  be  separated  by  filtration 
in  the  feed  pipe.  On  the  whole,  for  high  pressures  a  small  sur- 
face condenser  with  auxiliary  jet  seems  best.  To  give  a  rapid 
circulation  to  the  water,  and  render  the  heating  surface  efficient 


380  POWER  AND   PERFORMANCE   OF   ENGINES. 

in  the  highest  degree,  the  tubes  should  he  upright  with  the  water 
within  them  ;  and  the  furnaces  should  be  fed  with  coal  by  self- 
acting  mechanism,  which  would  abridge  the  labor  of  firing,  and 
insure  the  work  being  better  done.  To  reduce  the  strain  on  the 
engine  at  the  beginning  of  the  stroke,  when  steam  of  a  high 
pressure  is  employed,  the  stroke  should  be  long,  the  piston  small 
in  diameter,  and  a  considerable  velocity  of  piston  should  be  em- 
ployed ;  or,  where  there  are  two  engines,  the  steam  may  be  ex- 
panded from  the  cylinder  of  one  engine  into  the  cylinder  of  the 
other  engine,  according  to  Nicholson's  system,  whereby  twice  the 
expansion  will  be  obtained  with  only  the  same  apparatus. 

Relative  surface  areas  of  Boilers  and  Condensers. — The 
evaporative  power  of  land  and  marine  boilers  per  square  foot  of 
heating  surface,  depends  very  much  upon  the  structure  and  con- 
figuration of  the  boiler.  In  some  marine  engines  a  performance 
of  six  times  the  nominal  power  has  been  obtained  with  a  propor- 
tion of  heating  surface  in  the  boiler  of  only  12  square  feet  per 
nominal  horse-power ;  and  as  about  half  of  this  power  was  ob- 
tained by  expanding  the  steam,  1  cubic  foot  of  water  was  evap- 
orated by  every  4  square  feet  of  heating  'surface,  which  is  a 
smaller  proportion  even  than  that  which  obtains  commonly  in 
locomotives.  In  such  cases  the  proportion  of  cooling  surface  in 
the  condenser  has  been  made  equal  to  the  amount  of  heating 
surface  in  the  boiler ;  and  the  amount  of  cooling  surface  in  the 
condenser  relatively  to  the  amount  of  the  heating  surface  of  the 
boiler  should  manifestly  have  reference  to  the  activity  of  that 
heating  surface.  So  in  like  manner  it  should  be  influenced  by 
the  amount  of  expansion  which  the  steam  undergoes  in  the  cyl- 
inder ;  since  the  steam,  in  communicating  power,  parts  with  a 
corresponding  quantity  of  heat.  A  still  more  important  condi- 
tion of  the  action  of  the  condenser  is,  that  the  water  shall  pass 
through  the  tubes  with  rapidity,  and  that  it  shall  flow  in  the  op- 
posite direction  to  the  steam,  so  that  the  hottest  steam  shall 
meet  the  warmest  water ;  as  warm  water  will  suffice  to  condense 
hot  steam,  which  would  be  quite  inoperative  in  condensing  at- 
tenuated vapour.  A  common  proportion  of  condenser  surface 
in  modern  engines  is  '15  that  of  the  boiler  surface.  Thus  a 


INTERNAL    CORROSION    OF    BOILERS.  381 

boiler  with  20  square  feet  of  heating  surface  will  have  15  square 
feet  of  heating  surface.  But  the  largest  part  of  this  surface  is 
required  to  obtain  the  last  pound  or  two  of  exhaustion ;  and  it 
is  preferable  to  employ  a  moderate  surface  to  condense  the  bulk 
of  the  steam,  and  to  condense  the  residual  vapour  by  a  small  jet 
of  salt  water  let  in  from  the  sea.  It  is  found  advisable  to  admit 
a  small  quantity  of  salt  water  on  other  grounds.  For  the  fresh 
water  in  the  boiler,  as  it  forms  no  scale,  leaves  the  boiler  subject 
to  the  corrosive  influence  produced  by  placing  a  mass  of  copper 
tubes — on  which  the  sea  water  acts  chemically — in  connexion 
with  the  mass  of  wet  iron  which  constitutes  the  boiler ;  and,  as 
in  Sir  Humphrey  Davy's  arrangement  for  protecting  copper 
sheathing  by  iron  blocks,  the  copper  tubes  are  protected  at  the 
expense  of  the  boiler,  since  the  communicating  pipes  and  the 
water  within  them  form  an  efficient  connexion.  It  would  be 
easy  to  break  the  circuit  so  far  as  the  metal  is  concerned  by  in- 
terposing glass  flanges  between  the  flanges  of  the  pipes.  But 
this  would  not  stop  the  communication  by  the  water  itself,  and 
the  best  course  appears  to  satisfy  the  corroding  conditions  by 
placing  blocks  of  zinc  within  the  condenser,  which  might  be 
corroded  instead  of  the  tubes  or  the  boiler.  The  present  anti- 
dote to  the  corrosive  action  consists  in  the  introduction  of  a  cer- 
tain proportion  of  salt  water  into  the  boiler,  which  is  intended 
to  shield  the  evaporating  surfaces  from  corrosive  action  by  de- 
positing a  coating  of  scale  upon  those  evaporating  surfaces. 
But  in  this  arrangement  we  have  necessarily  an  excess  of  water 
entering  the  boiler ;  for  we  have  not  only  all  the  water  returned 
which  passes  off  as  steam,  but  a  certain  proportion  of  sea  water 
besides.  It  will  consequently  be  necessary  to  provide  for  the 
excess  being  blown  out  of  the  boiler ;  and  the  question  is,  whether, 
as  we  must  introduce  such  an  arrangement,  it  would  not  be  ad- 
visable, with  low  pressures,  to  make  the  proportions  such  as 
would  enable  us  to  dispense  with  the  surface  condenser  alto- 
gether ?  If  it  is  retained  at  all,  it  should  only  be  retained  in 
such  shorn  proportions  as  to  condense  the  grossest  part  of  the 
steam — the  water  resulting  from  which  should  be  sent  into  the 
boiler  quite  hot,  and  the  rarer  part  of  the  steam  should  be  con- 


382      POWER  AND  PERFORMANCE  OF  ENGINES. 

densed  by  a  jet  of  salt  water  of  about  the  same  dimensions  as 
that  already  employed.  It  is  very  necessary  to  be  careful  in  the 
case  of  surface  condensers  to  prevent  any  leakage  of  air,  which, 
if  mingled  with  the  steam,  would  form  a  wall  of  air  against  the 
refrigeratory  surface,  which  would  prevent  the  contact  of  the 
steam  and  hinder  the  condensation,  precisely  as  it  was  found  to 
do  in  the  old  engines  of  Newcomen,  where  air  was  purposely 
admitted  to  form  a  stratum  between  the  hot  steam  and  the  cold 
cylinder ;  and  which  diminished  the  loss  from  the  condensation 
of  the  steam  within  the  cylinder  to  a  very  material  extent. 

Example  of  modern  marine  engine  and  boiler. — As  an  exam- 
ple of  the  proportions  of  marine  engines  and  boilers  and  con- 
densers of  approved  modern  construction,  I  may  here  recapitu- 
late the  main  particulars  of  the  machinery  of  the  screw  steamer 
'  Rhone,'  constructed  for  the  West  India  Mail  Company  by  the 
Millwall  Iron  Company  in  1865. 

These  engines  are  on  the  inverted  cylinder  principle  of  500 
horse-power.  There  are  two  cylinders  of  72  inches  diameter 
and  4  feet  stroke,  and  the  estimated  number  of  revolutions  per 
minute  is  52.  The  cylinders  are  supported  on  massive  hollow 
standards  resting  on  a  bed  plate  of  the  same  construction.  There 
are  two  air-pumps  wrought  by  links  and  levers  from  two  pins  on 
the  ends  of  the  piston  rods.  The  surface  condenser  is  placed 
between  the  two  air-pumps,  and  is  fitted  with  brass  tubes  placed 
horizontally,  and  resting  in  vertical  tube  plates.  The  two  end 
plates  have  screwed  stuffing  boxes,  with  cotton  washer  packing 
for  each  tube.  The  tubes  are  divided  into  three  groups  or  sec- 
tions, through  each  of  which  the  condensing  wftter  successively 
passes ;  and  the  water  enters  from  the  lower  end  of  the  con- 
denser and  escapes  at  the  upper  end,  where  the  steam  enters,  so 
that  the  hottest  water  meets  the  hottest  steam.  The  two  circu- 
lating pumps  are  placed  opposite  each  other,  and  are  wrought  by 
a  crank  on  the  end  of  the  crank  shaft.  The  steam  is  condensed 
outside  the  tubes ;  and  the  condensed  water  flows  down  to  the^ 
air-pumps,  by  which  it  is  pumped  to  the  hot  well,  and  from 
which  it  is  taken  to  the  boilers  in  the  usual  way. 

The  crank  shaft  is  of  Krupp's  cast  steel  in  two  pieces, 


PRINCIPLE  OP  GIFFARD'S  INJECTOR.  383 

coupled  by  flanges.  The  screw  shaft  is  of  iron,  covered  with 
brass  in  the  stern  tube,  and  working  in  lignum  vitse  bearings  in 
the  stern  tube  and  after  stern  post.  The  boilers  are  in  four 
separate  parts,  and  fitted  with  a  superheating  apparatus  consist- 
ing of  a  series  of  vertical  iron  tubes  4J-  inches  bore,  on  the  plan 
of  Mr.  Ritchie,  the  company's  superintending  engineer. 

The  surface  condenser  has  3,566  tubes,  £  inches  external 
diameter,  and  9  feet  2J  inches  long  between  the  tube  plates. 
The  surface  of  the  tubes  is  6,525  square  feet,  or  13-05  square  feet 
per  nominal  horse-power.  The  two  circulating  pumps  are  dou- 
ble acting  25"  diameter,  with  a  trunk  of  17"  diameter  on  one 
end  of  the  plungers.  The  boilers  have  20  furnaces  3'  0£"  wide, 
with  fire  bars  of  6  feet  8  inches  in  length.  The  total  area  of 
fire  grate  is  400  square  feet,  =  0'8  square  feet  per  nominal  horse- 
power. The  number  of  brass  tubes  in  the  boiler  is  1,180  of  3£ 
external  diameter  and  6  feet  8  inches  long.  The  total  heating 
surface  in  the  boilers  is  9,800  square  feet,  or  19*6  square  feet  per 
nominal  horse-power.  In  the  superheater  the  surface  is  2,160 
square  feet,  or  4*32  square  feet  per  nominal  horse-power,  making 
the  total  heating  surface  in  boiler  and  superheater  23*92  square 
feet  per  nominal  horse-power.  The  area  of  heating  surface  in 
the  boiler  per  square  foot  of  grate  is  24'5  square  feet,  and  the 
area  of  superheating  surface  per  square  foot  of  grate  is  5-4  square 
feet,  making  the  total  heating  surface  in  boiler  and  superheater 
29'9  square  feet  per  square  foot  of  grate.  The  total  area  of  the 
condenser  surface  is  '66  of  the  total  heating  surface  in  the  boiler, 
and  '54  of  the  total  area  of  the  heating  surface  of  boiler  and 
superheater  taken  together.  These  engines  are  very  strong,  and 
manifestly  embody  the  results  of  the  long  experience  of  steam 
navigation  which  the  West  India  Mail  Company  must  now  pos- 
sess. The  workmanship  and  materials  are  of  the  very  first 
quality;  and  accurate  adjustment  and  conscientious  construc- 
tion are  manifested  throughout. 

GiffarcCs  Injector. — This  instrument,  which  feeds  boilers  by 
a  jet  of  steam  discharged  into  the  feed  pipe,  acts  on  the  principle 
that  the  particles  of  water  which  obtain  a  high  velocity  when 
they  flow  oxit  as  steam  retain  this  velocity  when  reduced  by 


384  POWER   AND    PERFORMANCE    OF   ENGINES. 

condensation  to  the  form  of  water ;  and  a  jet  of  water  of  great 
velocity  is  capable  of  balancing  a  correspondingly  high  head,  or 
a  pressure  greater  than  that  which  subsists  within  the  boiler. 
The  jet  consequently  penetrates  the  boiler,  as  we  can  easily  un- 
derstand any  jet  would  do  which  has  a  greater  velocity  than  a 
similar  jet  escaping  from  the  boiler.  These  injectors,  though 
very  generally  employed  in  locomotives,  are  not  much  used  for 
land  or  marine  boilers ;  and  in  their  present  form  they  occasion 
much  waste,  as  the  steam  by  which  they  are  actuated  is  drawn 
from  the  boiler,  whereas  it  ought  to  be  the  steam,  or  a  portion 
of  it,  which  escapes  to  the  condenser  or  the  atmosphere.  These 
injectors,  like  Bourdon's  gauges,  and  other  instruments  employed 
in  the  steam-engine,  are  not  made  by  engineers,  but  are  a  dis- 
tinct manufacture;  and  the  manufacturers,  on  being  supplied 
with  the  necessary  particulars,  furnish  the  proper  size  of  instru- 
ment in  each  particular  case.  The  proper  diameter  of  the  nar- 
rowest part  of  the  instrument  to  deliver  into  the  boiler  any  given 
number  of  gallons  per  hour,  may  be  found  by  dividing  the  num- 
ber of  gallons  required  to  be  delivered  per  hour  by  the  square 
root  of  the  pressure  of  the  steam  in  atmospheres,  and  extracting 
the  square  root  of  the  quotient,  which,  multiplied  by  the  con- 
stant number  '0158,  gives  the  diameter  in  inches  at  the  smallest 
part.  Contrariwise,  if  we  have  the  size,  and  wish  to  find  the 
delivery,  we  multiply  the  constant  number  63'4  by  the  diameter 
in  inches  and  square  the  product,  which,  multiplied  by  the  square 
root  of  the  pressure  of  the  steam  in  atmospheres,  gives  the  de- 
livery in  gallons  per  hour.  These  rules  correspond  very  closely 
with  the  tables  of  the  deliveries  of  different  sizes  published  by  the 
manufacturers,  Messrs.  Sharp,  Stewart,  and  Co.,  of  Manchester. 

POWEE  BEQUIBED   TO   PEBFOEM  VAEIOtTS   KINDS   OP   WORK. 

The  power  required  to  obtain  any  given  speed  in  a  given 
steamer  will  be  so  fully  discussed  in  the  next  chapter  that  the 
subject  need  not  be  further  referred  to  here ;  and  in  my  '  Cate- 
chism of  the  Steam-Engine '  I  have  recapitulated  the  amount  of 
power,  or  the  size  of  engine,  required  to  thrash  and  grind  corn, 
spin  cotton,  work  sugar  and  saw  mills,  press  cotton,  drive  piles, 


EFFICIENCY    OF    HYDRAULIC    MACHINES.  385 

dredge  earth,  and  blow  furnaces.  The  subject,  however,  is  so 
important  that  I  shall  here  recapitulate  other  cases  for  the  most 
part  derived  from  experiments  made  with  the  dynamometer  in 
France  by  General  Morin,*  whose  researches  on  this  subject 
have  been  highly  interesting,  and  have  been  conducted  with 
much  care  and  ability. 

Comparative  efficiency  of  different  machines  for  rawing  water. 
— Of  the  different  pumps  experimented  upon  by  General  Morin, 
the  result  of  eight  experiments  made  with  pumps  draining  mines 
showed  that  the  effect  utilised  was  66  per  cent,  of  the  power 
expended.  But  in  these  cases  there  was  considerable  loss  from 
leakage  from  the  pipes.  At  the  salt  works  of  Dreuze  the  useful 
effect  was  52-3  per  cent,  of  the  power  expended.  In  fire-engine 
pumps  employed  to  deliver  the  water  pumped  at  a  height  of 
from  12  to  20  feet,  the  proportion  of  the  water  delivered  to  the 
capacity  of  the  pump  was,  in  the  pumps  of  the  following  makers 
— Merryweather,  Tylor,  Perry,  Carl-Metz,  Letestu,  Flaud,  and 
Perrin,  respectively,  as  follows :— '920,  '887,  '910,  '974,  -910,  -920, 
and  '900  ;  while  the  percentage  of  useful  effect  relatively  with 
the  power  expended  was  39-7,  39-1,  30-2,  28-7,  27'1,  19'4,  and 
15 -5,  respectively.  "With  a  higher  pressure,  the  efficiency  of  the 
whole  of  the  pumps  increased ;  and  when  employed  in  throwing 
water  with  a  spout-pipe  the  delivery  of  water  relatively  with 
the  effective  capacity,  or  space  described  by  the  piston,  was,  when 
the  names  are  arranged,  as  follows : — Carl-Metz,  Merryweather, 
Tylor,  Letestu,  Perry,  Flaud,  Perrin,  and  Lamoine,  respective- 
ly, -950,  -810,  -565,  -870,  '910,  -912,  '950,  and  '900 ;  while  the 
proportion  of  useful  effect,  or  percentage  of  work  done  relatively 
with  the  power  expended,  was  80,  57'3,  54'5,  45<2,  37'8,  33-4, 
28'8,  and  17'5,  in  the  respective  cases.  In  the  membrane  pump 
of  M.  Brule  the  efficacy  was  found  to  be  40  to  45  per  cent,  of  the 
power  expended.  In  the  water- works  pumps  of  Ivry,  construct- 
ed by  Cave,  the  efficiency  was  found  to  be  53  per  cent,  of  the 
power  expended ;  and  in  the  water- works  of  St.  Ouen,  by  the 
same  maker,  76  per  cent.  It  is  desirable  that  the  buckets  of 
the  pumps  of  water-works  should  move  slowly,  otherwise  the 
*  Aide  Mfmoire,  by  General  Morin,  5th  edition,  1864. 

17 


386      POWER  AND  PERFORMANCE  OF  ENGINES. 

water  will  go  off  with  considerable  velocity,  involving  a  corre- 
sponding loss  of  power.  The  area  through  the  valves  should  be 
half  the  area  of  the  pump,  and  the  area  of  the  suction  and  forcing 
pipes  ought  to  be  equal  to  three-fourths  of  the  area  of  the  body 
of  the  pump.  Waste  spaces  should  be  avoided.  The  loss  of  water 
through  the  valves  before  they  shut  is,  in  good  pumps,  about  10 
per  cent. 

In  a  chain-pump  the  efficiency  was  found  to  be  38  per  cent., 
but  in  many  chain-pumps  the  efficiency  is  much  more  than  this. 
The  efficiency  of  the  Persian  wheel  was  found  to  increase  very 
much  with  the  height  to  which  the  water  was  raised.  For 
heights  of  1  yard  it  was  48  per  cent.,  for  2  yards  57,  for  3  yards  63, 
for  4  yards  66,  and  for  6  yards  and  upwards  70  per  cent,  of  the 
power  consumed.  For  a  wheel  of  pots  the  efficiency  is  60  per 
cent. ;  Archimedes  screw,  65  per  cent. ;  scoop  wheel  with  flat 
boards  moving  in  a  circular  channel,  70  per  cent.;  improved 
bucket  wheel,  82  per  cent.,  and  tympan-wheel,  or,  as  it  is  some- 
tunes  called,  "Wirtz's  Zurich  machine,  88  per  cent.  This  machine 
should  dip  at  least  a  foot  into  the  water  to  give  the  best  results. 
In  the  belt-pump  the  efficiency  was  found  to  be  43  per  cent.;  in 
Appold's  centrifugal  pump,  65  per  cent. ;  in  the  centrifugal 
pump,  with  inclined  vanes,  42  per  cent.,  and  with  radial  vanes, 
24  per  cent.  In  Gwynn's  pump  the  efficiency  was  30  per  cent. 

In  the  Archimedes  screw  the  diameter  is  usually  one-twelfth 
of  the  length,  and  the  diameter  of  the  newel  or  central  drum 
should  be  one-third  of  the  diameter  of  the  screw.  It  ought  to 
have  at  least  three  convolutions,  and  the  line  traced  by  the 
screw  on  the  enveloping  cylinder  should  have  an  angle  of  67°  to 
70°  with  the  axis.  The  axis  itself  should  make  an  angle  of  from 
30°  to  45°  with  the  horizon.  There  is  a  sensible  advantage  ob- 
tained from  working  hand-pumps  by  a  crank  instead  of  a  lever. 

Old,  French  Flour  Mill  at  Senelle. — Diameter  of  millstones, 
70  inches ;  number  of  revolutions  per  minute,  70 ;  quantity  of 
corn  ground  and  sifted  per  hour,  260'7  Ibs. ;  power  consumed, 
3-34  horses.  The  power  is  in  all  cases  the  power  actually  exert- 
ed, as  ascertained  by  the  dynamometer. 

English  Flour  Mill  near  Metz. — Diameter  of  millstones,  5118 


POWER   REQUIRED   TO   DRIVE   MILLS.  387 

inches;  number  of  revolutions  per  minute,  110;  weight  of  mill- 
stones, 1  ton ;  corn  ground  per  hour  by  each  pair,  220  Ibs. ;  with 
two  pairs  of  millstones  acting,  one  bolting  machine  and  one  win- 
nowing machine,  the  power  consumed  was  8^  horse-power. 

English  Flour  Mill  near  Verdun, — Diameter  of  millstones, 
51'18  inches;  number  of  revolutions  per  minute,  110;  quantity 
of  corn  ground  per  hour  by  each  pair,  or  by  each  revolving  mill- 
stone, 220  Ibs. ;  with  two  stones  revolving  the  power  consumed 
was  5 "64  horses.  The  power  consumed  by  one  winnowing  ma- 
chine and  two  bolting  machines,  with  brushes  sifting  1,650  Ibs. 
of  flour  per  hour,  was  6  £  horses.  In  another  mill  the  number  of 
turns  of  the  millstone  was  486  per  minute,  the  quantity  of  corn 
ground  by  each  horse-power  was  120  Ibs.,  and  the  quantity  of 
corn  ground  per  hour  was  110  Ibs.  of  which  72-7  per  cent,  was 
flour,  7' 8  per  cent,  was  meal,  and  19 -5  per  cent,  was  bran.  In  a 
portable  flour-mill,  with  machinery  for  cleaning  and  sifting,  the 
total  weight  was  1,000  Ibs. 

Barley  Mill. — Number  of  revolutions  of  the  millstone  per 
minute,  246;  barley  ground  per  hour,  143-68  Ibs. ;  motive  force  ill 
horses,  3-11 ;  barley  ground  per  hour  by  each  horse-power,  48-2 
Ibs.  The  products  were,  of  first  and  second  quality  of  barley 
flour,  60-12  per  cent.,  of  meal  and  bran,  30*25  per  cent.,  and  of  . 
bran  and  waste,  9-63  per  cent. 

Eye  Mill. — Number  of  revolutions  of  the  millstone  per  minute, 
448 ;  rye  ground  per  hour,  92-114  Ibs. ;  power  expended,  2-86 
horses;  temperature  of  flour,  60*8°  Fahr. ;  products  64-9  per 
cent,  of  flour,  9-l  per  cent,  of  meal,  and  26  per  cent,  of  bran.  In 
another  rye  mill  the  revolutions  of  the  millstones  per  minute 
were  232  ;  rye  ground  per  hour,  180  Ibs.  by  2-19  horse-power, 
and  the  rye  ground  per  hour  by  each  horse-power  was  82-21 
Ibs.  The  products  were  72*5  per  cent,  of  flour;  17'5  per  cent, 
of  meal  and  fine  bran,  and  10  per  cent,  of  bran  and  waste. 

Maize  Mill. — Number  of  revolutions  of  the  millstone  per 
minute,  246 ;  maize  ground  per  hour,  73-96  Ibs. ;  motive  force  in 
horses,  2'69  ;  maize  ground  per  hour  by  each  horse-power,  27'5 
Ibs.  Products:  first  and  second  quality  of  flour,  61-1  per  cent. ; 
meal  and  fine  bran,  30-2  per  cent. ;  bran  and  waste,  4-7  per  cen% 


388      POWER  AND  PERFORMANCE  OF  ENGINES. 

Vermicelli  Manufactory.— External  diameter  of  edge  runners, 
66'93  inches;  internal  diameter  of  edge  runners,  62'99  inches; 
number  of  revolutions  of  the  arbour  of  the  mill  per  minute,  4 ; 
pounds  of  paste  prepared  per  hour,  77  Ibs. ;  power  expended,  2'95 
horse-power. 

Sean  Mill. — Number  of  revolutions  of  the  millstone  per  min- 
ute, 496;  power  expended  per  hour,  1'76  horse. 

Oil  Mill. — "Weight  of  edge  runners,  6,600  Ibs. ;  number  of 
turns  of  the  vertical  spindle  per  minute,  6  ;  weight  of  seed  intro- 
duced every  ten  minutes,  55  Ibs. ;  weight  of  seed  crushed  daily, 
3,300  Ibs. ;  product  in  oil  in  12  hours,  1,320  Jbs. ;  power  expend- 
ed, 2-72  horses. 

S'aw  Mill — "Weight  of  the  saw  frame,  842'6  Ibs.  When  cut- 
ting dry  oak  8.73  inches  thick,  with  1  blade  in  operation,  the 
reciprocations  or  strokes  of  the  saw  were,  88  per  minute,  the 
surface  cut,  -525  square  foot,  and  the  power  expended  3'3  horses. 
When  cutting  the  same  wood  with  4  blades  in  operation,  the 
number  of  strokes  of  the  saw  per  minute  was  79 ;  the  surface  cut  by 
each  per  minute  '433  square  foot,  or  1'73  square  foot  per  minute 
for  the  4;  and  the  power  expended  was  3*70  horses,  which  is 
equivalent  to  28  square  feet  cut  per  hour  by  1  horse-power.  When 
cutting  four-year  seasoned  oak,  12-4  inches  thick,  with  4  blades, 
making  90  strokes  per  minute,  the  surface  cut  by  each  blade  was 
•35  square  foot,  and  the  surface  cut  by  the  4  blades,  1/41  square 
foot.  When  the  saw  was  run  along  the  middle  of  a  cylindrical 
log  of  beech  one-year  cut,  23'6  inches  diameter,  the  number  of 
strokes  of  the  saw  per  minute  was  88 ;  the  surface  cut  per  minute, 
"968  square  foot ;  and  the  power  expended,  3  horses.  In  these 
experiments  the  breadth  of  the  saw  cut  was  '157  inch,  and  the 
experiments  show  that  it  does  not  take  more  power  to  drive  a 
frame  with  one  saw  than  to  drive  a  frame  with  four,  the  great- 
est part  of  the  power  indeed  being  consumed  in  giving  motion  to 
the  frame.  The  common  estimate  in  modern  saw  mills,  when 
the  frame  is  filled  with  saws,  is,  that  to  cut  45  superficial  feet  of 
pine,  or  34  of  oak  per  hour,  requires  1  indicated  horse-power. 
The  crank,  which  moves  the  frame  up  and  down,  and  which  is 
usually  placed  in  a  pit  under  tho  machine,  .should  have  balance 


POWER   REQUIRED   TO   DRIVE    SAWS.  389 

weights  applied  to  it,  the  momentum  of  which  weights,  when 
the  saw  is  in  action,  will  be  equal  to  that  of  the  reciprocating 
frame.  In  some  cases  the  weight  of  the  saw  frame  is  borne  by 
a  vacuum  cylinder,  and  with  a  20-inch  stroke  it  makes  120 
strokes  per  minute. 

Circular  Saw. — Diameter  of  saw,  27'5  inches ;  thickness  of 
oak  cut,  8*73  inches;  number  of  revolutions  per  minute,  266; 
surface  cut  per  minute,  T93  square  foot;  power  consumed  3*55 
horses.  When  set  to  cut  planks  of  dry  fir,  10'62  inches  broad, 
and  one  inch  thick,  the  number  of  revolutions  made  by  the  saw 
per  minute  was  244;  surface  cut  per  minute,  7'67  square  feet; 
and  the  power  expended,  7'35  horses.  These  results  show  that 
in  sawing  the  smaller  class  of  timber  one  circular  saw  will  do  at 
least  as  much  work  as  four  reciprocating  saws,  with  the  same 
expenditure  of  power.  The  surface  cut  is,  in  all  these  cases,  under- 
stood to  be  the  height  multiplied  by  the  length,  and  not  the  sum 
of  the  two  faces  separated  by  the  saw.  The  speed  of  the  circular 
saw  here  given  is  not  half  as  great  as  that  now  commonly  em- 
ployed. Circular  saws  now  work  with  a  velocity  at  the  peri- 
phery of  6,000  to  7,000  feet  per  minute,  and  band  saws  with  a 
velocity  of  2,500  feet  per  minute,  and  it  is  generally  reckoned 
that  75  superficial  feet  of  pine,  or  58  of  oak,  will  be  sawn  per 
hour  by  a  circular  saw  for  each  indicated  horse-power  expended. 
Planing  machine  cutters  move  with  a  velocity  at  the  cutting  edge 
of  4,000  to  6,000  feet  per  minute,  and  the  planed  surface  travels 
forward  ^Oth  of  an  inch  for  each  cut. 

Reciprocating  Veneer  Saw. — Length  of  stroke  of  saw,  47'24 
inches;  thickness  of  the  blade,  '01299  inch;  breadth  of  saw  cut, 
•02562  inch;  length  of  teeth  for  mahogany  and  other  valuable 
woods,  -196  inch;  pitch  of  the  teeth,  -3939  inch;  distance  ad- 
vanced by  the  wood  each  stroke,  '0196  to  '03937  inch ;  number 
of  strokes  of  the  saw  per  minute,  180 ;  surface  cut  per  hour 
counting  both  faces,  107*64  square  feet ;  power  expended  0'66 
horses. 

Sawing  Machine  for  Stones. — Soft  sandstone :  breadth  of  saw- 
cot,  £  inch ;  time  employed  to  saw  10  square  feet,  5  minutes  25 
seconds ;  power  expended  4'54  horses.  Hard  sandstone :  breadth 


390  POWER   AND   PERFORMANCE    OP    ENGINES. 

of  saw-cut,  J  inch;  time  employed  to  cut  10  square  feet,  1  hour 
3V  minutes ;  power  expended  2  horses. 

Sugar  Mill  for  Canes. — A  three-cylinder  mill,  with  rollers 
5£  feet  long,  30  inches  diameter,  and  making  2J  turns  a  minute, 
driven  hy  an  engine  of  25  to  30  horse-power,  will  express  the 
juice  out  of  130  tons  of  canes  in  12  to  15  hours.  An  acre  of 
land  produces  from  10  to  20  tons  of  canes,  according  to  the  age 
and  locality  of  the  canes.  The  juice  stands  at  8°  to  12°  of  the 
saccharometer,  according  to  the  locality.  The  product  in  sugar 
varies  from  6  to  10  per  cent,  of  the  weight  of  the  canes,  accord- 
ing to  the  locality  and  mode  of  manufacture.  Well-constructed 
mills  give  in  juice  from  60  to  70  per  cent,  of  the  weight  of  the 
canes,  and  one  main  condition  of  efficiency  is,  that  the  rollers 
shall  travel  slowly,  as  with  too  great  a  speed  the  juice  has  not 
time  to  separate  itself  from  the  Avoody  refuse  of  the  cane,  and 
much  of  it  is  reabsorhed.  To  defecate  330  gallons  of  juice  6 
boiling-pans,  or  caldrons,  are  required,  4  scum  presses,  and  10 
filters ;  and  to  granulate  the  sugar  2  vacuum  pans,  6£  feet  diam- 
eter, are  required,  with  2  condensers,  and  it  is  better  also  to 
have  2  air-pumps.  The  steam  for  boiling  the  liquor  in  the 
vacuum  pans  is  generated  in  three  cylindrical  boilers,  each  6  feet 
in  diameter.  To  whiten  the  sugar  there  are  10  centrifugal 
machines,  driven  by  a  12-horse  engine,  which  also  drives  a  pair 
of  crushing-rollers.  The  sugar  in  the  centrifugal  machines  is 
wetted  with  syrup,  which  is  driven  off  at  the  circumference  of 
the  revolving  cylinders  of  wire  gauze,  carrying  with  it  most  of 
the  colouring  matter  of  the  sugar,  which  to  a  great  extent  adheres 
to  the  outside  of  the  crystals,  instead  of  being  incorporated  in  them, 
and  may  consequently  be  washed  off.  When  the  sugar  is  thus 
cleansed  it  is  again  dissolved,  and  the  syrup  is  passed  through 
deep  filters  of  animal  charcoal.  Provision  must  be  made  to 
wash  the  charcoal,  both  by  steam  and  by  water,  and  two  fur- 
naces, to  re-burn  the  animal  charcoal,  will  be  required. 

The  action  of  animal  charcoal  in  bleaching  sugar  is  not  well 
understood.  But  it  appears  to  be  due  to  certain  metallic  bases 
in  the  bones,  which  by  burning  are  brought  to  or  towards  the 
metallic  state,  from  the  superior  affinity  of  the  carbon  present 


EXAMPLE   OF   A   COTTON    MILL.  391 

for  the  oxygen  in  the  base  at  the  high  temperature  at  which  the 
re-burning  takes  place.  When,  however,  the  charcoal  is  mixed 
with  the  syrup,  the  metallic  base  endeavours  to  recover  the 
oxygen  it  has  lost,  by  decomposing  the  water,  leaving  thereby  a 
certain  quantity  of  hydrogen  in  the  nascent  state ;  and  this  hy- 
drogen appears  to  dissolve  the  small  particles  of  carbon  in  the 
sugar  which  detract  from  its  whiteness,  and  to  form  therewith 
a  colourless  compound.  When  the  metallic  basis  has  recovered 
all  its  lost  oxygen  the  charcoal  ceases  to  act,  and  has  to  be  re- 
burned  ;  and,  after  numerous  re-burnings,  the  charcoal  appears 
to  be  all  burned  out  of  the  bones,  when  re-burning  ceases  to  be 
of  service.  But  their  efficacy  might  be  restored  by  mingling  por- 
tions of  wood  charcoal.  The  use  of  charcoal  in  sugar  refining  is 
not  merely  a  source  of  expense  in  itself,  but  it  occasions  a  loss 
of  sugar,  as,  when  the  mass  of  charcoal  becomes  effete,  it  is  left 
saturated  with  syrup,  and  the  water  with  which  it  is  washed  has 
to  be  boiled  down,  to  recover  the  sugar  as  far  as  possible.  I 
consequently  proposed  several  years  ago  a  method  of  revivifying 
the  charcoal  without  removing  it  from  the  filter.  But  the 
method  has  not  yet  been  practically  adopted. 

The  begass,  or  woody  refuse  of  the  cane,  is  usually  employed 
to  generate  the  steam  in  the  boilers.  But  it  is  generally  neces- 
sary to  use  coal  besides. 

Fans  for  blowing  Air. — The  indicated  power  required  to 
work  a  fan  may  be  ascertained  by  multiplying  the  square  of  the 
velocity  of  the  tips  in  feet  per  second  by  the  collective  areas  of 
the  escape  orifices  in  square  inches,  and  by  the  pressure  of  the 
blast  in  pounds  per  square  inch,  and  finally  dividing  the  product 
by  the  constant  number  62,500,  which  gives  the  indicated  power 
required.  The  pressure  in  pounds  per  square  inch  may  be  de- 
termined by  dividing  the  square  of  the  velocity  of  the  tips  in  feet 
per  second  by  the  constant  number  97,300. 

Cotton-spinning  Mill. — Number  of  spindles,  26,000 ;  power 
consumed,  110  horses;  Nos.  of  yarn  spun,  30  to  40;  spindles 
with  preparation  driven  by  each  horse,  237.  It  is  reckoned  that 
each  machine  requires  1  horse-power. 

Another  example  of  a  Cotton  Mill. — Number  of  spindles, 


392  POWER   AND    PERFORMANCE    OF    ENGINES. 

14,508 ;  power  required  to  drive  them,  50'5  horses;  ISTos.  of  yarn 
spun,  30  to  40  ;  spindles  and  preparation  driven  by  each  horse- 
power, 287. 

Another  example  of  a  Cotton  Mill. — Number  of  spindles, 
10,476;  Nos.  of  yarn  spun,  30  to  40;  spindles  and  preparation 
driven  by  each  horse-power,  235. 

Details  of  power  required  l>y  each  Machine  in  Cotton  Mills. — 
One  beater  making  1,100  revolutions  per  minute,  with  ventilat- 
ing fan  making  half  this  number  of  revolutions,  cleaning  132  Ibs. 
of  cotton  per  hour,  requires  2'916  horse-power.  One  beater 
making  1,200  revolutions  per  minute,  with  combing  drum  1*23 
feet  diameter  and  2'8  feet  long,  making  800  revolutions  per  min- 
ute, and  preparing  132  Ibs.  of  cotton  per  hour,  requires  800  revo- 
lutions per  minute  and  1*767  horse-power.  Power  required  to 
work  the  fluted  cylinders  and  endless  web  of  this  machine,  '812 
horse.  Twelve  double-casing  cylinders,  with  eccentrics,  re- 
quiring 2*697  horses,  including  the  transmission  of  the  motion, 
or  per  machine,  '225  horse.  Transmitting  the  motion  for  26 
carding-machines  requires  1*82  horse-power.  One  simple  card, 
consisting  of  a  drum  39'37  inches  diameter  and  19*68  inches 
long,  making  130  revolutions  per  minute,  and  carding  2  Ibs.  of 
cotton  per  hour,  requires  '066  horse-power,  without  reckoning 
the  power  consumed  in  communicating  the  motion.  The  same 
card  working  empty  requires  '044  horse-power.  One  double- 
carding  machine  carding  4*18  Ibs.  of  cotton  per  hour,  requires 
•207  horse-power.  A  drawing-frame  drawing  119  Ibs.  per  hour 
requires  1'835  horse-power.  A  roving-frame,  with  60  spindles, 
with  cards,  making  525  revolutions  per  minute,  and  producing 
42  Ibs.  of  No.  7  rovings  per  hour,  requires  '760  horse-power. 
One  frame  with  screw-gearing,  having  60  spindles,  making  550 
revolutions  per  minute,  and  producing  42  Ibs.  of  No.  7  per  hour, 
requires  '486  horse-power.  Two  frames  with  screw-gearing, 
each  containing  96  spindles,  making  in  one  case  510  revolutions 
and  the  other  500  revolutions  per  minute,  producing  28*6  Ibs. 
of  No.  2*75  to  3  per  hour,  requires  1*482  horse-power.  Two 
frames  with  screw-gearing,  one  containing  78  spindles  making 
344  revolutions  per  minute,  and  the  other  60  spindles  making  260 


POWER   REQUIRED    TO   DRIVE   WOOLLEN    MILLS.       393 

revolutions  per  minute,  and  producing  57'2  Ibs.  of  No.  8  per 
hour,  requiring  '797  horse-power.  One  spinning-frame,  with 
cards,  having  240  spindles,  making  5,000  revolutions  per  minute, 
and  producing  T65  Ib.  of  yarn  of  No.  38  to  ]STo.  40  per  hour, 
requires  '686  horse-power,  and  in  another  experiment  '648  horse- 
power. Three  spinning-frames  for  weft,  having  each  360  spin- 
dles, making  4,840  revolutions  per  minute,  and  producing  8  Ibs. 
of  No.  30  to  No.  40  yarn  per  hour,  require  2*103  horse-power. 
One  retwisting  machine,  with  120  spindles,  making  3,000  revo- 
lutions per  minute,  requires  1'19  horse-power.  One  dressing 
machine  for  calico  35£  inches  wide,  with  ventilator :  speed  of 
the  principal  arbor,  176  revolutions  per  minute;  speed  of  the 
brushes,  45  strokes  per  minute ;  power  required,  '735  horse. 
The  same  machine,  with  the  ventilator  not  going,  requires  -206 
horse-power. 

Power-loom  Weaving. — To  drive  one  power-loom  weaving 
calico  35J  inches  wide,  and  82  to  90  picks  per  inch,  making  105 
strokes  per  minute,  requires,  taking  an  average  of  four  experi- 
ments, '1195  horse-power. 

Another  example  of  Power-loom  Weaving. — Number  of 
looms  weaving  calico  driven  by  water-wheel,  260  ;  dressing  ma- 
chines, 15 ;  winding  machines,  5  ;  warping  machines,  8  ;  small 
pumps,  6  ;  yards  of  calico  produced  per  month,  283,392  ;  power 
required  to  drive  the  mill,  25'6  horses;  number  of  looms,  with 
accessories,  moved  by  1  horse,  12. 

Another  example  of  Power-loom  Weaving. — Total  number  of 
looms,  60 ;  dressing  machines,  5 ;  warping  machines,  3 ;  wind- 
ing machines,  2;  monthly  production  of  cotton  cloth  called 
'Normandy  linen,'  47i  inches  wide,  360  pieces,  each  396  yards 
long ;  power  consumed,  8  horses ;  looms  with  their  accessories 
moved  by  each  horse-power,  7'8. 

Wool-spinning  Mill. — Machines  driven :  simple  cards,  29  ; 
double  cards,  2 ;  scribbling  beater,  1 ;  mules  of  240  spindles,  8 ; 
mules  of  200  spindles,  4;  lathes,  3;  power  consumed,  9'76 
horses.  Also  in  another  experiment  with  9  simple  and  3  double- 
carding  machines,  2  beaters,  and  2  scribbling  machines,  the 
power  consumed  in  driving  was  3*5  horses. 
17* 


394      POWER  AND  PERFORMANCE  OF  ENGINES. 

Another  example  of  a  Wool-spinning  Mill. — A  wheel  exert- 
ing 10  horse-power  drives  6  mules  of  240  spindles,  6  of  180,  2 
of  192,  2  of  120,  and  5  of  100,  making  in  all  3,644  spindles  ;  also 
32  carding  and  2  scribbling  machines.  Another  wheel,  also  ex- 
erting 10  horse-power,  drives  8  mules  of  240  spindles,  4  of  120, 
and  7  of  180,  making  in  all  3,660  spindles;  also  31  carding  and 
2  scribbling  machines,  and  2  beaters.  The  spindles,  number- 
ing in  all  7,304,  make  5,000  revolutions  per  minute,  and  the 
cards  88  to  89,  requiring  a  horse-power  for  365  spindles.  Prod- 
uct per  day  of  12  hours,  1,100  Ibs.  of  yarn  from  No.  12  to 
No.  13. 

Details  of  Power  consumed  in  spinning  Wool. — One  winding 
machine  with  16  bobbins,  without  counting  the  power  expended 
in  the  transmission  of  the  motion,  requires  to  drive  it  '259 
horse ;  3  winding  machines  with  64  bobbins  in  ah1,  with  power 
lost  by  transmission,  1'427  horse;  one  mule  spinning  No.  6  warp 
yarn,  with  220  spindles,  making  3,650  revolutions  per  minute, 
•259  horse.  One  mule  called '  Box-organ,'  spinning  No.  50  warp 
yarn  with  300  spindles,  making  3,200  revolutions  per  minute, 
requires  1*273  horse-power. 

Mill  for  spinning  Wool  and  weaving  Merinos. — Nineteen 
machines  to  prepare  the  combed  wool,  having  together  350 
rollers;  16  mules  with  3,400  spindles ;  one  winding  machine  of 
60  rollers  to  prepare  the  warp ;  2  warping  machines ;  2  self- 
acting  feeders;  100  power-looms;  2  lathes  for  wood  and  iron, 
and  1  pump,  require  in  all  30  horse-power.  Produce :  13,600 
cops  of  woollen  thread,  of  45  cops  to  the  lb.,  each  measuring 
792  yards.  The  looms  make  115  revolutions  per  minute,  and 
produce  daily  4  pieces  of  double- width  merino  of  68  yards  each, 
and  4  pieces  of  simple  merino  of  1*2  to  1-4  yard  broad,  and  each 
88  yards  long. 

Fulling  Mill. — In  falling  the  cloths  called  'Beauchamps,' 
each  piece  being  220  yards  long,  and  -66  yard  wide,  and  weigh- 
ing from  121  to  127  Ibs.,  the  fuller  making  100  to  120  strokes 
per  minute,  each  piece  requires  2  hours  to  full  it,  and  the  expen- 
diture of  2  horse-power  during  that  time. 

Flax  Manufacture. — A  machine  for  retting  the  flax,  having 


POWER   REQUIRED   TO   DRIVE   FLAX   MILLS.  395 

15  pairs  of  rollers  with  triangular  grooves,  requires  3'376  horse- 
power, and  the  heckles  '057  horse-power. 

One  fly  breaking-card  12-59  inches  diameter  and  47'24  inches 
long,  making  915  revolutions  per  minute,  with  a  drum  of  42'12 
inches  diameter,  and  47'24  inches  long,  making  76  revolutions 
per  minute ;  4  distributing  rollers,  having  a  diameter  of  4  inches 
and  a  length  of  47'24  inches,  making  380  revolutions  per  minute  ; 

3  travellers,  5  inches  diameter  and  47'24  inches  long,  making  10 
turns  per  minute,  and  one  combing  cylinder  15  inches  diameter 
and  47'24  inches  long,  making  6  revolutions  per  minute,  require 
together  1-939  horse-power,  and  produce  17  Ibs.  of  carded  flax 
per  hour. 

One  finishing  carding  cylinder,  40  inches  diameter  and  47'24 
inches  long,  making  176  revolutions  per  minute ;  5  distributing 
rollers,  4  inches  diameter,  making  23  revolutions  per  minute ; 

4  travellers,  5  inches  diameter,  making  7'3  revolutions  per  min- 
ute ;  1  combing  cylinder,  15  inches  diameter,  making  3 -4  revolu- 
tions per  minute,  together  require  '811  horse-power,  and  produce 
8£  Ibs.  of  carded  flax  per  hour. 

One  spinning-machine,  containing  132  spindles,  making  2,700 
revolutions  per  minute,  spinning  yarns  from  No.  7  to  No.  9,  re- 
quires 1*24  horse-power,  and  produces  3f  Ibs.  of  yarn  per  hour. 

One  spinning-machine,  having  168  spindles,  making  2,700 
revolutions  per  minute,  and  producing  3  Ibs.  of  No.  18  to  24 
yarn  per  hour,  requires  1'96  horse-power. 

Wet  spinning  of  flax :  one  drawing-frame  drawing  a  sliver 
for  No.  20  yarn,  requires  -493  horse;  drawing-frame  drawing 
sliver  for  No.  50  yarn,  requires  *487  horse ;  drawing-frame  draw- 
ing sliver  for  No.  70  yarn,  requires  *495  horse. 

Second  drawing-frame,  drawing  two  slivers  for  yarns  Nos. 
20  and  30,  requires  '68  horse ;  second  drawing-frame,  drawing 
two  slivers  for  yarns  Nos.  30  and  40,  requires  -544  horse ;  second 
drawing-frame,  drawing  one  sliver  for  No.  60  yarn  and  one  for 
No.  70,  requires  '617  horse. 

Third  drawing-frame,  drawing  two  slivers  for  yarns  Nos.  30 
to  60,  requires  '69  iorse. 

Koving-frame  of  8  spindles,  preparing  the  flax  for  yarn  No. 


396  POWER   AND    PERFORMANCE    OF    ENGINES. 

20,  requires  '608  horse;  roving-frame  of  8  spindles,  preparing 
the  flax  for  No.  30  yarn,  requires  '486  horse;  frame  of  16  spin- 
dles, preparing  the  flax  for  No.  40  yarn,  requires  '987  horse- 
power. 

Paper  Manufacture. — In  some  cases  the  pulp,  or  stuff  of 
which  paper  is  made,  is  obtained  by  beating  the  rags  by  stamp- 
ers ;  but  more  generally  it  is  produced  by  placing  the  rags  be- 
tween revolving  cylinders  stuck  full  of  knives.  When  produced 
by  stampers,  the  proportions  of  the  apparatus  are  as  follows : 
•weight  of  stampers,  220  Ibs. ;  distance  of  the  centre  of  gravity 
from  the  axis  of  rotation,  4  feet ;  rise  of  the  centre  of  gravity 
each  stroke,  3|  inches;  number  of  stampers,  16;  number  of 
lifts  of  each  stamper  per  minute,  55  ;  weight  of  rags  pounded  in 
12  hours  by  each  stamper,  33  Ibs. ;  weight  of  stuff  produced  in 
12  hours  by  each  stamper,  122  Ibs. ;  power  consumed,  2'7  horses. 

Chopping-cylinders,  for  preparing  the  pulp :  number  of  cyl- 
inders working,  2 ;  number  of  turns  of  cylinders  per  minute, 
220 ;  weight  of  rags  chopped  and  purified  in  12  hours,  528  Ibs. ; 
power  consumed,  4'48  horses. 

In  another  instance,  10  cylinders  for  preparing  the  pulp, 
making  200  revolutions  per  minute,  1  paper-making  machine, 
cutting-machines,  pump,  and  accessories,  consumed  50-horse 
power.  The  machine  made  13  yards  of  paper  per  minute,  and 
the  produce  was  1  ton  of  printing  paper  per  day  of  24  hours. 

In  another  instance,  28  pulping-cylinders,  and  3  paper-mak- 
ing machines  produced  2  to  3  tons  of  paper  per  day  of  24  hours, 
and  consumed  113  horse-power. 

Printing  Machinery. — Printing  large  numbers  is  now  per- 
formed by  cylindrical  stereotype  plates,  revolving  continuously ; 
and  the  '  Tunes '  and  other  newspapers  of  large  circulation  are 
thus  printed.  The  impressions  are  taken  from  the  types  in 
papier  mache,  and  in  twenty  minutes  a  large  stereotype  plate 
is  ready  to  be  worked  from.  The  power  required  to  drive 
this  machine  varies  with  the  number  of  impressions  required  in 
the  hour.  For  5,000  impressions  per  hour,  the  power  required 
is  3-75  horses ;  for  6,000  impressions,  4*77  horses;  7,000  impres- 
sions, 5-9  horses;  8,000  impressions,  7'03 horses;  9,000  impres- 


WEAVING   BY   COMPRESSED    AIR.  397 

sions,  8'75  horses;  and  10,000  impressions,  10-35  horses.  The 
paper  should  be  supplied  to  such  machines  in  a  continuous  web, 
with  a  cutter  to  cut  off  the  sheets  at  the  proper  intervals,  and  a 
steam  cylinder  to  dry  and  press  them.  But  this  has  not  yet 
been  done.  The  machine  could  also  be  easily  made  to  perforate 
the  paper  along  the  edges  of  the  leaves,  and  to  fold  each  paper 
up  and  put  a  printed  and  stamped  paper  envelope  around  it,  so 
as  to  be  ready  at  once  to  put  into  the  post-office  or  to  distribute 
by  hand.  The  most  expeditious  mode  of  stereotyping  would 
be  to  use  steel  types  set  on  a  cylinder,  against  which  another 
cylinder  of  type-metal  is  pressed,  and  the  paper  would  then  be 
printed  in  the  same  manner  as  calico. 

Glass  Works. — Mill  to  grind  red  lead :  to  grind  3  tons,  the 
vertical  arbor  requires  to  make  for  the  first  ton  20  revolutions 
per  minute,  for  the  second  25,  and  for  the  third  40,  consuming 
5'28  horse-power.  Vertical  millstones,  to  grind  clay  and  broken 
crucibles;  diameter  of  the  granite  stones  or  runners,  3'V  feet; 
thickness,  1/4  foot;  weight,  1  ton;  distance  of  edge  runners 
from  central  spindle,  4  feet ;  number  of  turns  of  the  arbor  per 
minute,  Tg-;  power  consumed  1'92  horse.  In  the  12  hours  6  or 
8  charges  of  about  300  Ibs.  each  of  old  glass  pots  are  ground, 
and  about  3  tons  of  dry  clay.  Wheels  for  cutting  the  glass,  1YO ; 
lathes  for  preparing  the  cutting  wheels,  5 ;  lathes  for  metal,  2 ; 
power  consumed,  17*9  horses;  wheels  driven  by  each  horse- 
power, 9 -5. 

Iron-  Worlcs. — The  weekly  yield  of  each  smelting  furnace  in 
"Wales  is  from  100  to  120  tons ;  pressure  of  blast,  2£  to  3  Ibs.  per 
square  inch ;  temperature  of  the  blast,  600°  Fahr. ;  yield  weekly 
of  each  refining-furnace,  80  to  100  tons;  of  each  puddling-furnace, 
18  tons;  of  each  balling-furnace  for  bars,  30  tons;  of  each  ball- 
ing-furnace for  rails,  80  tons ;  iron  rolled  weekly  by  puddle  rolls, 
300  tons ;  by  rail  rolls,  600  tons ;  power  required  to  work  each 
train  of  rail  rolls,  250  horses ;  to  work  puddle  rolls  and  squeezer, 
80  horses;  small  bar  train,  60  horses;  pumping  air  into  each 
blast-furnace,  60  horses ;  into  each  refining-furnace,  26  horses ; 
rail  saw,  12  horses. 

Weaving  ty  compressed  air. — In  common  power-looms,  the 


398  POWER   AND    PERFORMANCE    OF   ENGINES. 

shuttle  is  driven  backward  and  forward  by  a  lever  which  imi- 
tates the  action  of  the  arm  in  the  hand-loom.  But  it  has  long 
been  obvious  to  myself  and  others  that  it  might  be  shot  back- 
ward and  forward  like  a  ball  out  of  a  gun,  by  means  of  com- 
pressed air.  This  innovation  has  now  been  practically  carried 
out.  But  the  benefits  derivable  from  the  practice  have  been 
much  exaggerated,  and  a  much  more  comprehensive  improve- 
ment than  this  is  now  required.  Indeed,  reciprocating  looms  of 
all  kinds  are  faulty,  as  they  make  much  noise,  consume  much 
power,  do  little  work,  and  cannot  be  driven  very  fast ;  and  the 
proper  remedy  lies  in  the  adoption  of  a  circular  loom  in  which 
the  cloth  will  be  woven  in  a  pipe,  and  in  which  many  threads 
of  weft  will  be  fed  in  at  the  same  time. 

Circular  Loom. — The  obvious  difficulty  in  a  circular  loom,  is 
to  drive  the  shuttle  round  continuously  within  the  walls  formed 
by  the  warp.  One  mode  of  driving  proposed  by  me,  is  by  mag- 
nets or  other  suitable  form  of  electro-motive  machine,  which 
does  not  require  contact ;  and  the  shuttle  should  be  a  circular 
ring,  with  many  cops  placed  in  it,  so  that  many  threads  might 
be  woven  in  at  once.  The  desideratum,  however,  is  to  weave  a 
vertical  pipe  with  the  bobbins  of  the  weft  in  the  centre  of  the 
circle ;  and  this  may  be  done  by  depositing  the  thread  between 
metallic  points,  like  circular  heckles,  which  points  will  change 
their  positions  inward  or  outward  at  each  time  a  thread  is  de- 
posited. These  points  would  conduct  the  threads  of  the  warp. 


CHAPTER  VIL 

STEAM  NAVIGATION. 

STEAM  navigation  embraces  two  main  topics  of  enquiry : — 
the  first,  what  the  configuration  of  a  vessel  shall  be  to  pass 
through  the  water  at  any  desired  speed  with  the  least  resist- 
ance ;  and  the  second,  what  shall  be  the  construction  of  ma- 
chinery that  shall  generate  and  utilise  the  propelling  power 
with  the  greatest  efficiency.  The  second  topic  has,  in  most  of 
its  details,  been  already  discussed  in  the  preceding  pages ;  and  it 
will  now  be  proper  to  offer  some  remarks  on  the  remaining 
portion  of  the  subject. 

The  resistance  of  vessels  passing  through  the  water  is  made 
up  of  two  parts : — the  one,  which  is  called  the  bow  and  stern 
resistance,  being  caused  partly  by  the  hydrostatic  pressure  forc- 
ing back  the  vessel,  arising  from  the  difference  of  level  between 
the  bow  and  stern,  and  partly  by  the  power  consumed  in  blunt 
bows  in  giving  a  direct  impulse  to  the  water ;  while  the  other 
part  of  the  resistance,  and  the  most  important  part,  is  that  due 
to  the  friction  of  the  water  on  the  sides  and  bottom  of  the  ship. 
The  bow  and  stern  resistance  may  be  reduced  to  any  desired 
extent  by  making  the  ends  sharper.  But  the  friction  of  the  bot- 
tom cannot  be  got  rid  of,  or  be  materially  reduced,  by  any  means 
yet  discovered. 

When  a  vessel  is  propelled  through  water,  the  water  at  the 
bow  has  to  be  moved  aside  to  enable  the  vessel  to  pass ;  and  the 
velocity  with  which  the  water  is  moved  sideways  will  depend 
upon  the  angle  of  the  bow  and  the  speed  of  the  vessel.  When 


400  STEAM    NAVIGATION. 

these  elements  are  known  it  is  easy  to  tell  with  what  velocity 
the  water  will  be  moved  aside ;  and  when  we  know  the  velocity 
with  which  the  water  is  moved,  we  can  easily  tell  the  power 
consumed  in  moving  it,  which  power  will,  in  fact,  he  the  weight 
of  the  water  moved  per  minute  multiplied  by  the  height  from 
which  a  body  must  fall  by  gravity  to  acquire  the  same  velocity. 
But  as  nearly  all  the  power  thus  consumed  in  moving  aside  the 
water  at  the  bow  of  a  vessel  is  afterwards  recovered  at  the  stern 
by  the  closing  in  of  the  water  upon  the  run,  it  is  needless  to  go 
into  this  investigation  further  than  to  determine  what  amount 
of  power  is  wasted  by  the  operation,  or  in  other  words,  what 
amount  of  power  is  expended  that  is  not  afterwards  recovered. 

If  the  vessel  to  be  propelled  is  of  a  proper  form,  each  particle 
of  water  will  be  moved  sideways  by  the  bow,  in  the  same  man- 
ner as  the  ball  of  a  pendulum  is  moved  sideways  by  gravity,  so 
as  to  enable  the  vessel  to  pass ;  and  when  the  broadest  part  of 
the  vessel  has  passed  through  the  channel  thus  created,  each 
particle  of  water  will  swing  backward  again  until  it  comes  to 
rest  at  the  stern.  There  will  be  no  waste  of  power  in  this 
operation,  except  that  incident  to  the  friction  of  the  moving 
water ;  just  as  in  the  swinging  of  a  pendulum  there  is  no  expen- 
diture of  power  beyond  that  which  is  necessary  to  overcome 
the  friction  of  the  air  upon  the  moving  ball.  But  as  the  move- 
ment of  the  vessel,  however  well  she  may  be  formed,  will  some- 
what raise  the  water  at  the  bow,  and  somewhat  depress  the 
water  at  the  stern,  there  will  be  a  certain  hydrostatic  pressure 
required  to  be  continually  overcome  as  the  vessel  advances  in 
her  course,  which  opposition  constitutes  the  bow  and  stern  re- 
sistance ;  and  this,  with  the  friction  of  the  bottom,  make  up  the 
whole  resistance  of  the  ship.  Before,  however,  proceeding  to 
investigate  the  amount  of  this  hydrostatic  resistance,  it  will  be 
proper  to  show  how  accidental  sources  of  loss  may  be  elim- 
inated from  the  problem  by  the  introduction  of  that  particular 
form  of  vessel  which  will  make  this  resistance  a  minimum ;  and 
I  will  therefore  first  proceed  to  indicate  in  what  way  such  form 
of  vessel  may  be  obtained. 

If  we  take  a  short  log  of  wood,  such  as  is  shown  by  the 


FORM    OF    MINIMUM   RESISTANCE.  401 

dotted  lines  A  B  c  r>  E  F  G,  in  the  annexed  figure  (fig.  41),  and  if 
we  proceed  to  enquire  in  what  way  we  shall  mould  this  log  into 
a  model  which  shall  offer  the  least  possible  hydrostatic  resist- 
ance in  being  drawn  through  the  water,  we  have  the  following 
considerations  to  guide  us  in  arriving  at  the  desired  knowledge : 
We  shall,  for  the  sake  of  simplification,  suppose  that  the  cross 
section  of  the  completed  model  is  to  be  rectangular,  or  in  other 
words,  that  the  model  is  to  have  vertical  sides  and  a  flat  bottom ; 
for  although  this  is  not  the  best  form  of  cross  section,  as  I  shall 

Fig.  41 


afterwards  show,  the  supposition  of  its  adoption  in  this  case  will 
simplify  the  required  explanations. 

"We  first  draw  a  centre  line  x  y  longitudinally  along  the  top 
of  the  model  from  end  to  end,  and  continue  the  line  vertically 
downward  at  the  ends  as  at  y  z,  which  vertical  lines  will  form 
the  stem  and  stern  post  of  the  model.  At  right  angles  to  the 
first  line,  and  at  the  middle  of  the  length  of  the  model,  we  draw 
the  line  a,  which  answers  to  the  midship  frame ;  and  midway 
between  a  and  the  ends  we  draw  other  two  lines  5  5.  We  may 
afterwards  draw  any  convenient  number  of  equi-distant  cross- 


402  STEAM    NAVIGATION. 

lines,  or  ordinates,  as  they  are  termed,  that  we  find  to  be  conven- 
ient. Now  as,  by  the  conditions  of  the  problem,  the  particles 
of  water  have  to  swing  sideways  like  a  pendulum,  in  order  that 
the  resistance  may  be  a  minimum,  the  particle  which  encounters 
the  stem  at  x  must  be  moved  sideways  very  slowly  at  first,  like  a 
heavy  body  moved  by  gravity,  but  gradually  accelerating  until 
it  arrives  at  5,  midway  between  x  and  a,  where  its  velocity  will 
be  greatest ;  and  this  point  answers  to  the  position  of  the  ball 
of  the  pendulum  when  it  has  reached  the  bottom  of  the  arc,  and 
has  consequently  attained  its  greatest  velocity.  Thereafter  the 
motion,  which  before  was  continually  accelerated,  must  be  now 
continually  retarded,  as  it  is  in  any  pendulum  that  is  ascending 
the  arc  in  which  it  beats,  or  in  any  ball  which  is  projected  up- 
wards into  the  air  against  the  force  of  gravity.  When  the  par- 
ticle of  water  has  attained  the  position  on  the  side  of  the  model 
which  is  opposite  to  the  midship  frame  #,  it  will  have  come  to 
rest,  this  being  the  point  answering  to  the  position  of  the  pen- 
dulum at  the  top  of  its  arc,  and  when  just  about  to  make  the 
return  beat.  Thereafter  the  particle  which  was  before  moved 
outwards,  will  now  move  inward  with  a  velocity,  slow  at  first, 
but  continually  accelerating,  until  it  attains  the  position  on  the 
side  of  the  model  which  is  opposite  to  the  frame  5,  when  the 
velocity  again  begins  to  diminish ;  and  the  particle  finally  comes 
to  rest  at  the  stern.  A  particle  of  water  that  is  moved  in  this 
way  will  be  moved  with  the  minimum  of  resistance ;  for  since 
it  retains  none  of  the  motion  in  it  that  has  been  imparted,  but 
surrenders  the  whole  gradually  without  impact  or  percussion, 
by  the  time  it  has  come  finally  to  rest,  there  can  be  no  power 
consumed  in  moving  it  except  that  due  to  friction  only.  "Wher- 
ever the  water  is  not  moved  in  this  manner  it  will  either  retain 
some  of  the  motion,  which  implies  a  corresponding  waste  of 
power,  or  heat  will  be  generated  by  impact,  which  also  involves 
a  corresponding  waste  of  power.  That  the  water  may  be  moved 
in  the  same  manner  as  a  pendulum  is  moved,  is  obviously  possi- 
ble, by  giving  the  proper  configuration  to  the  sides  of  the 
model ;  and  in  fact,  if  an  endless  sheet  of  paper  be  made  to 
travel  vertically  behind  a  pendulum,  with  a  pencil  or  paint 


CURVE   OF   GRAVITY. 


403 


brush  stuck  in  the  ball,  the  proper  form  for  the  side  of  the 
model  will  be  marked  upon  the  paper.  The  curve,  however, 
which  is  a  parabolic  one,  may  be  described  geometrically  as 
follows : — 

If  we  compute  the  height  through  which  a  heavy  body  falls 
by  gravity  in  any  given  number  of  seconds,  we  shall  find  that  in 
the  first  quarter  of  a  second  it  will  have  fallen  through  lTg^  foot, 
in  the  second  quarter  of  a  second  3^  feet,  in  the  third  9|,  in  the 
fourth  16TL,  in  the  fifth  25^,  in  the  sixth  36T3F,  in  the  seventh 
49T4592,  in  the  eighth  64^,  in  the  ninth  81f  J,  and  in  the  tenth 
quarter  of  a  second  100||.  The  height  fallen  through,  there- 
fore, or  the  space  described  by  a  falling  body  in  a  given  time, 


varies  as  the  square  of  the  time  of  falling ;  and  any  body  which 
is  to  be  moved  in  the  same  manner  as  a  falling  body  is  moved  by 
gravity,  must  have  the  motion  imparted  to  it  gradually  at  the 
same  rate  of  progression.  If,  then,  we  draw  a  line,  x  y  in  fig. 
42,  and  which  line  we  may  suppose  to  be  the  vertical  plane  of  the 
keel,  then  if  we  form  the  parallelogram  A  B  o  D,  with  the  line  x  y 
passing  through  the  middle  of  it,  and  make  this  parallelogram  one- 
fourth  of  the  length  of  the  vessel  and  half  the  breadth,  and  divide 
the  line  x  y  into  any  number  of  convenient  parts  or  ordinates, 
say  10,  by  the  vertical  co-ordinates  numbered  from  1  to  10,  then 
if  we  cause  the  lengths  of  these  successive  and  equidistant  co- 
ordinates, measuring  from  the  line  x  y,  to  follow  the  same  law 
of  increase  that  answers  to  the  height  through  which  a  body 


404:  STEAM   NAVIGATION. 

falls  by  gravity  in  successive  and  equal  portions  of  time,  a  line 
traced  through  the  ends  of  these  different  lines  will  give  the 
right  form  for  the  side  of  a  vessel  to  have,  in  order  that  it  may 
move  the  water  sideways,  in  the  same  manner,  or  according  to 
the  same  law,  by  which  a  heavy  body  falls  vertically  by  gravity ; 
and  consequently  such  line  is  the  proper  water-line  of  a  ship  form- 
ed under  the  conditions  supposed,  in  order  that  it  may  have  a. 
minimum  resistance.  The  heights  of  the  several  vertical  ordi- 
nates — which  are  drawn  on  a  different  scale  from  the  lengths, 
marked  on  the  line  x  y,  are — 1,  4,  9,  16,  25,  36,  49,  64,  81,  and 
100,  which,  it  will  be  seen,  are  the  squares  of  the  horizontal 
ordinates  1,  2,  3,  4,  5,  6,  7,  8,  9,  and  10 ;  and  the  scale  by 
which  these  vertical  ordinates  are  measured  is  formed  by  divid- 
ing the  distance  y  D,  which  represents  one-fourth  of  the  breadth 
of  the  vessel,  into  100  equal  parts.  The  ordinate  y  D  is  therefore 
equal  to  100  of  those  parts,  the  next  ordinate  to  81  of  them,  the 
next  to  64  of  them,  and  so  on,  until  the  height  vanishes  at  * 
altogether.  We  might  have  divided  the  line  x  y  into  nine  equal 
parts,  or  into  8,  or  7,  or  any  other  convenient  number.  In  such 
case  the  vertical  line  y  D  would  have  to  be  divided  into  81  equal 
parts  to  obtain  the  vertical  scale,  or  into  64,  or  into  49,  according 
as  9,  8,  or  7  had  been  the  number  selected ;  but  the  number  of  parts 
into  which  y  D  is  divided  must  always  be  equal  to  the  square  of 
the  number  of  ordinates,  or  the  square  of  the  number  of  parts 
into  which  the  horizontal  line  is  divided.  As  it  is  difficult  to 
measure  the  hundredth  part  of  such  a  small  length  as  y  D,  we 
may  call  the  number  of  parts  10  instead  of  100,  in  which  case 
the  length  of  the  next  ordinate  will  be  8*1,  of  the  next  6'4,  of  the 
next  4*9,  and  so  on — the  whole  of  the  squares  being  divided  by  10 ; 
which  proceeding  will  in  no  way  affect  the  result,  as,  in  point  of 
fact,  the  difference  is  only  much  the  same  thing  as  if  we  meas- 
ured in  inches  instead  of  in  feet. 

In  the  figure,  x  y  is  five  times  longer  than  y  D,  and  x  y  rep- 
resents one-fourth  of  the  length  of  the  vessel,  and  y  D  one-fourth 
of  the  breadth.  The  curved  line  x  D  represents  the  proper  form 
of  the  water-line  of  the  front  half  of  the  fore  body  in  the  case  of 
a  vessel  of  these  proportions,  and  with  a  rectangular  cross-sec- 


CURVE    OF    GRAVITY. 


405 


tion.  The  water-line  of  the  second  half  of  the  fore  body  is 
formed  by  repeating  the  same  curve,  but  inverted  and  reversed, 
this  will  be  made  obvious  by  an  inspection  of  fig.  43,  where  the 
first  half  of  the  fore  body  is  repeated  on  a  smaller  scale ;  and  the 
second  portion  of  the  fore  body  is  added  thereto,  thus  continuing 
the  water-line  to  the  midship  frame  a  a.  Here  the  rectangle 
enclosing  the  water-line  of  the  first  half  of  the  vessel  is  shown  in 
dotted  lines,  as  is  also  the  rectangle  enclosing  the  water-line  of 
the  first  half  of  the  fore  body ;  and  it  is  plain  that  the  shaded 
space  a  d  is  the  exact  duplicate  of  the  shaded  space  x  d;  so  that 
if  the  figure  x  d  has  been  obtained,  we  may  obtain  the  figure  d  a 
by  cutting  out  of  the  paper  the  figure  x  d,  inverting  it  and  re- 
Fig.  43. 


versing  it,  so  that  the  line  x  d  shall  coincide  with  the  line  d  a,  and 
the  point  x  with  the  point  a ;  or  the  figure  d  a  may  be  con- 
structed by  co-ordinates  in  exactly  the  same  manner  as  the  figure 
x  d.  If  the  vertical  sides  of  the  vessel  be  formed  with  the  curve 
shown  by  the  curve  line  x  a,  then  it  will  follow  that  a  particle 
of  water  encountering  the  stem  at  as,  will  be  moved  aside  slowly 
at  first,  and  with  a  rate  continually  increasing,  like  a  body  falling 
by  gravity,  until  the  frame  5  lying  midway  between  the  stem  and 
the  midship  frame  is  reached,  at  which  point  the  water  will  be 
moving  sideways  with  its  greatest  velocity.  Thereafter  the  vessel 
will  not  move  the  water,  but  merely  follow  up  the  motion  already 
given  to  it,  and  as  the  water,  when  no  longer  impelled  sideways 
by  the  vessel,  will  move  slower  and  slower,  and  gradually  come  to 
rest,  so  the  vessel  will  have  less  and  less  following  up  to  do,  until 
at  the  midship  frame  a  a,  the  side  motion  of  the  water  ceases 


406  STEAM   NAVIGATION. 

altogether.  Thereafter  the  water  begins  to  move  in  the  opposite 
direction  to  fill  up  the  vacuity  at  the  stern  left  by  the  progress 
of  the  vessel.  The  water  gravitates  into  the  run  slowly  at  first, 
and  the  velocity  increases  until  the  point  midway  between  the 
midship  frame  and  the  stern  is  attained,  at  which  point  the  ve- 
locity is  greatest ;  and  from  thence  the  velocity  of  the  water, 
flowing  inward,  continually  diminishes,  until  it  comes  to  rest  at 
the  stern. 

A  rectangular  box,  such  as  that  shown  by  the  dotted  lines 
A  B  o  D  E  F  G,  fig.  41,  into  which  the  model  exactly  fits,  is  called 
its  circumscribing  parallelepiped ;  and  it  will  be  at  once  appar- 
ent, on  a  reference  to  fig.  41,  that  the  bulk  or  capacity  of  the 
model  is  exactly  one-half  of  its  circumscribing  parallelepiped. 
The  rectangle  x  d  is  equal  to  the  rectangle  d  y,  and  the  shaded 
space  *  d  being  equal  to  the  shaded  space  d  a,  the  area  included 
between  the  water-line  and  the  vertical  plane  of  the  keel,  namely, 
the  area  x  y  a,  is  clearly  equal  to  the  rectangle  d  d  y  a.  But  that 
rectangle,  and  the  rectangle  standing  beneath  it,  are  equal  to  the 
whole  area  within  the  water-line  of  the  fore  body,  and  two 
similar  rectangles  are  equal  to  the  area  within  the  water-line  of 
the  after  body.  As  these  four  rectangles  form  just  half  the  area 
of  the  circumscribing  parallelogram,  the  total  area  within  the 
water-line  is  equal  to  half  the  area  of  the  circumscribing  parallel- 
ogram. But  the  area  multiplied  by  the  depth  gives  the  capacity, 
and  as  the  depth  of  the  model  is  the  same  as  that  of  the  box,  or 
circumscribing  parallelepiped,  while  the  area  of  the  circumscrib- 
ing parallelogram  is  twice  that  of  the  area  of  the  figure  within 
the  water  line,  it  follows  that  the  volume  or  bulk  of  the  model  is 
just  one-half  of  the  circumscribing  parallelepiped.  This  forms  a 
measure  of  sharpness  which  in  no  case  it  is  useful  to  exceed,  if 
the  section  be  made  rectangular,  or,  in  other  words,  if  the  vessel 
be  built  with  a  flat  bottom  and  vertical  sides.  But  if  the  vessel 
be  built  with  a  rising  floor  the  effect  is  equivalent  to  a  reduction 
of  the  breadth,  and  the  circumscribing  parallelepiped  would,  in 
such  case,  be  that  answering  to  the  equivalent  breadth.  What- 
ever be  the  form  of  the  cross-section,  however,  the  sectional  area 
at  each  successive  frame  should  be  equal  to  that  of  a  vessel  with 


FISHES   CONFORM    TO   THE   LAW.  407 

a  rectangular  section  having  water-lines  formed  on  the  principle 
which  has  been  here  explained.  There  are  other  curves,  no 
doubt,  which  equally  with  that  described  by  a  pendulum  fulfil  the 
indication  of  beginning  and  terminating  the  motion  gradually  so 
as  to  involve  no  loss  of  power,  and  any  of  these  curves  are  eligi- 
ble as  the  water-line  of  a  ship.  But  the  pendulum  curve  is  the 
most  readily  understood,  and  the  most  conveniently  applicable 
to  practical  uses,  while  it  perfectly  fulfis  the  required  indica- 
tions. If  in  any  intended  vessel  we  have  a  given  form  of  cross- 
section,  and  a  given  ratio  of  length  to  breadth,  we  can  easily 
determine  the  proper  water-lines  of  such  a  vessel  by  taking  the 
case  of  a  hypothetical  vessel  of  rectangular  cross-section  having 

Fig.  M. 


the  same  area  of  midship-section,  and  by  forming  the  water-lines 
for  this  hypothetical  vessel  on  the  principle  already  explained. 
The  area  of  cross-section  at  each  successive  frame  of  this  hypo- 
thetical vessel,  will  be  the  proper  area  at  each  successive  frame 
of  the  intended  vessel.  It  is  obvious  that,  according  to  the  prin- 
ciple here  unfolded,  the  form  of  water-line  must  vary  with 
every  alteration  of  the  cross -section ;  and  in  some  cases,  although 
the  same  rate  of  displacement  as  that  already  indicated  is  pre- 
served, the  water-lines  will  cease  to  be  hollow  at  any  part.  Thus 
the  cylindrical  solid,  with  pointed  ends,  shown  in  fig.  44,  is  virtu- 
ally of  the  same  form  as  that  represented  in  fig.  41,  since  the 
area  of  each  successive  circular  cross-section  is  the  same  as 
those  of  each  rectangular  cross-section  in  fig.  41.  This  solid  is 


408  STEAM   NAVIGATION. 

supposed  to  be  wholly  immersed.  It  has,  in  some  cases,  been 
made  an  objection  to  the  use  of  hollow  water-lines  for  ships,  that 
in  the  case  of  fishes,  however  fast  swimming,  no  hollow  lines  are 
to  be  found  in  them.  Fig.  44,  however,  which  resembles  the 
form  of  a  fish,  shows  that  fishes  form  no  exception  to  the  appli- 
cation of  the  law  of  progressive  parabolic  displacement  already 
explained ;  and  if  a  fast-swimming  fish  be  cut  across  at  equal 
distances,  and  the  areas  of  these  sections  be  computed  and  laid 
down  with  a  rectangular  outline  of  uniform  depth,  it  will  be 
found  that  the  skin  or  covering  placed  over  the  ends  of  these 
sections  or  frames  will  assume  the  very  form  which  has  been  de- 
lineated in  the  foregoing  figures  as  that  proper  for  a  solid  in- 
tended to  pass  through  the  water  with  the  least  amount  of  hydro- 
static resistance. 

In  fig.  44,  x  y  is  the  axis  of  the  pointed  cylindrical  solid ;  and 
a  is  the  circle  or  section  which  answers  to  the  midship  frame, 
and  5  5  the  sections  answering  to  the  frames  lying  midway  be- 
tween the  centre  frame  and  the  ends.  The  other  lines  corre- 
sponding to  those  marked  on  the  model  shown  in  fig.  41,  and  the 
area  of  each  successive  circle  is  equal  to  the  area  of  each  successive 
rectangular  section  of  the  model  delineated  in  fig.  41.  The 
water  consequently  will  be  displaced  at  the  same  rate  by  one 
solid  as  by  the  other.  For  actual  vessels,  with  rounded  bilges 
and  more  or  less  rise  of  floor,  the  form  of  the  water-lines  will 
be  neither  that  shown  in  fig.  41  nor  fig.  44,  but  will  be  some- 
thing intermediate  between  the  two ;  but  such,  nevertheless, 
that  the  transverse  sectional  area  of  that  part  of  the  vessel 
beneath  the  water-line  shall  at  each  successive  frame  vary  in  the 
ratio  pointed  out. 

As  water  is  practically  incompressible  by  any  force  which  a 
ship  can  bring  to  bear  upon  it,  the  water  which  a  ship  displaces 
must  find  some  outlet  to  escape ;  and  it  will  escape  in  the  line 
of  least  resistance,  which  is  to  the  surface.  A  particle  of  water, 
therefore,  on  which  a  ship  impinges,  will  have  two  kinds  of  mo- 
tion— one  a  motion  outwards  and  inwards,  such  as  has  been 
already  described  as  resembling  the  motion  of  a  pendulum,  and 
the  other  a  motion  upwards  and  downwards,  caused  by  the  ne- 


BEST   FORM   OF   CROSS-SECTION. 


409 


cessity  of  the  particles  beneath  the  surface  rising  up  towards  the 
surface  to  allow  the  vessel  to  pass,  and  afterwards  of  sinking 
down  at  the  stern  to  fill  the  vacuity  which  the  progress  of  the 
vessel  would  otherwise  occasion.  This  last  motion  also  resem- 
hles  that  of  a  pendulum,  the  particles  of  water  at  the  stem  rising 
up  until  they  attain  their  greatest  height  at  the  midship  frame, 
and  then  again  subsiding  towards  the  stern. 

It  is  not  difficult,  from  these  considerations,  to  deduce  the 
conclusion  that  the  form  of  vessel  with  a  flat  floor  is  not  the  best 
which  can  be  adopted,  as  will  be  more  clearly  understood  by  a 
reference  to  fig.  45,  where  the  rectangle  D  E  F  a,  represents  the 

Fig.  45. 


cross-section  beneath  the  water-line  of  a  flat-floored  vessel  at  the 
point  midway  between  the  stem  and  the  midship  frame,  while 
the  triangle  ABO  is  the  cross-section  of  a  sharp-floored  vessel 
at  the  same  point,  and  with  the  same  sectional  area.  The 
draught  of  water  in  each  case  is  10  feet,  represented  by  the 
figures  1  to  10 ;  and  the  half  breadth  of  the  vessel  with  the 
rectangular  cross-section  at  this  point  of  the  length  is  5  feet,  which 
also  is  one-fourth  of  the  midship  breadth.  As  the  water  has  to 
be  set  back  from  the  line  of  the  stem  to  the  line  of  the  side,  or 
in  the  case  of  the  flat-floored  vessel,  through  a  distance  of  5  feet, 
we  may  represent  the  power  consumed  in  the  operation  by  5 
feet  multiplied  by  the  mean  hydrostatic  pressure  of  the  water  <5n 
each  square  foot.  The  mechanical  power  required  to  be  ex- 
18 


410  STEAM  NAVIGATION. 

pended  therefore  in  separating  the  water  in  the  two  sections  will 
be  as  follows : — 

Rectangular  section.  Triangular  section. 

5x1=5  9x1=9 

5x2  =   10  8x2  =   16 

6x3  =  15  7x3  =  21 

5x4  =  20  6x4  =  24 

5x5  =  25  5x5  =  25 

5x6  =  30  4x6  =  24 

5x7  =  35  3x7  =  21 

5x8  =  40  2x8  =  16 

5x9  =  45  1x9=9 

5    xlO  =  50  0   xlO  =     0 


275  165 

The  area  of  the  triangle  ABC  being  equal  to  that  of  the 
rectangle  DBF  G,  the  weight  of  water  displaced  by  a  foot  in  the 
length  of  the  vessel  will  be  the  same  whichever  form  of  cross- 
section  is  adopted  ;  and  as  the  areas  of  the  shaded  triangles  A  D  x 
and  B  r>  a;,  or  of  the  corresponding  triangles  B  F  x  and  o  G  a,  are  also 
the  same,  they  represent  equal  amounts  of  outward  motion  of 
the  water,  and  also  equal  amounts  of  displacement.  In  the  one 
case,  however,  this  motion  is  produced  against  a  much  greater 
hydrostatic  pressure  than  in  the  other  case  ;  and  as  by  shifting 
the  triangle  B  F  a;  into  the  position  o  G  x  —  whereby  we  enable  the 
vessel  to  move  outward  the  same  volume  of  water,  but  against  a 
less  hydrostatic  resistance  —  we  transform  the  rectangle  H  B  F  G 
into  the  triangle  H  B  o,  it  follows  that  there  is  less  resistance 
caused  by  the  movement  of  the  water  in  the  case  of  triangular 
cross  sections  than  in  the  case  of  rectangular.  The  rubbing  sur- 
face too  is  less  in  the  triangular  section.  By  the  principles  of 
geometry,  applicable  to  all  right-angled  triangles,*  (BF)S  +  (F  *)2= 


*  This  is  proved  by  the  47th  Proposition  of  the  first  book 
of  Euclid,  which  shows  that  the  area  of  the  square  described 
on  the  side  A  o,  opposite  to  the  right  angle  of  a  right-angled 
triangle  is  equal  to  the  sum  of  the  squares  described  on  the 
other  sides  A  B  and  B  c. 


BEST    FOEM    OF    CROSS-SECTION. 


411 


(B  a1)'2.  As  B  F  =  5  feet  and  F  x  also  =  5  feet,  then  (B  F)-  =  25, 
and  (F  x)2  =  25,  and  25  +  25  =  50,  consequently  B  2  =  ^/50  =  7 
nearly.  The  length  of  the  immersed  triangular  outline  is  conse- 
quently 7  x  4  —  28  feet,  whereas  the  length  of  the  rectangular 
outline  =  3  x  10  =  30  feet.  As  the  resistance  due  to  the  friction 
of  the  bottom  varies  as  the  quantity  of  rubbing  surface,  it  follows 
that,  as  regards  friction,  the  triangular  outline  is  also  the  more 
eligible.  Instead,  however,  of  a  simple  triangle,  it  is  preferable 

Fig.  46. 


that  the  cross-section  should  be  of  the  order  of  figure  indicated 
as  the  best  for  the  horizontal  water-lines;  and  the  same  con- 
siderations which  led  to  the  conclusion  that  this  form  would 
offer  the  least  resistance  in  the  case  of  a  body  moving  through 
stationary  water  lead  also  to  the  conclusion  that  it  will  offer  the 
least  resistance  to  water  moving  upwards  past  a  stationary  ob- 
ject— which  a  ship  may  be  supposed  to  be  relatively  to  the  plane 
in  which  she  floats.  Such  a  figure  is  represented  in  fig.  46,  in 
which  the  triangular  section  is  shown  in  dotted  lines,  and  the 
waving  lines  pass  alternately  without  and  within  the  dotted  lines. 
The  cross-section  of  the  vessel  is  for  the  most  part  of  the  outline 


412 


STEAM   NAVIGATION. 


a  semi-circle  m  m  m — a  semicircle  being  the  form  which  presents 
the  smallest  perimeter  relatively  with  the  immersed  sectional 
area ;  but  the  triangular  portion  in  n  is  added  both  to  prevent 
the  vessel  from  rolling  inconveniently,  and  to  bring  the  outline 
into  the  waving  curve  which  other  considerations  point  out  as 
the  most  eligible.  One  of  these  considerations,  as  already  men- 
tioned, is  that  it  best  fulfils  the  condition  of  beginning  the  up- 
ward displacement  slowly,  and  another  is  that  it  eifects  the  least 
possible  alteration'  in  the  shape  of  the  displaced  water.  In 

Fig.  47. 


altering  the  form  of  a  liquid,  as  in  altering  the  form  of  a  solid, 
there  is  a  certain  expenditure  of  force ;  and  although  this  ex- 
penditure in  the  case  of  a  liquid  is  relatively  very  small,  it  is 
large  enough  to  be  worthy  of  attention  in  a  case  where  large 
amounts  are  consumed  in  giving  motion  to  water.  It  hence  be- 
comes better,  since  the  displaced  fluids  must  assume  the  form  of 
a  wave,  to  effect  the  displacement  so  that  this  form  shall  be  at 
once  acquired,  instead  of  some  other  form  being  first  given  to  it 
which  is  subsequently  changed  by  the  action  of  other  forces. 
This  reasoning  will  be  better  understood  by  a  reference  to  fig. 


BEST   FORM   OF   CROSS-SECTION.  413 

47,  where  w  L  is  the  water-level,  m  n  the  cross-section  of  half  the 
vessel,  and  A  A  the  wave  which  would  be  raised  if  there  were 
no  outward  motion  of  the  water,  but  only  an  upward  motion. 
The  outward  motion  reduces  the  altitude  to  some  such  small 
elevation  as  a  a.  Nevertheless  it  is  advisable  that  the  outline  of 
the  wave  a  a  should  be  the  same  order  of  figure  as  the  outline 
of  the  wave  A  A,  only  laterally  extended.  Such  indeed  is  the 
shape  it  will  necessarily  assume ;  and  there  will  be  less  change 
of  shape  and  therefore  less  motion  of  the  internal  particles,  if  the 
wave  a  a  is  drawn  out  sideways  from  a  block  of  water  of  the 
form  A  A,  than  if  drawn  out  from  a  rectangular,  triangular,  or  any 
other  form  of  block.  The  dotted  lines  indicate  the  directions  in 
which  the  pressure  will  be  transmitted,  and  if  we  suppose  these 
lines  to  be  tubes,  it  will  be  obvious  that  the  surface  of  the  water 
in  these  tubes  will  only  conform  to  the  outline  of  a  wave,  if  the 
side  of  the  vessel  has  that  outline.  If  we  suppose  the  portions 
of  those  tubes  rising  above  the  water-line  to  be  very  much  en- 
larged, then  the  height  of  the  outline  will  fall  from  A  A  to  a  a, 
but  the  same  order  of  figure  will  still  be  preserved,  as  it  involves 
less  expenditure  of  power  to  give  this  form  at  once  than  to  give 
some  other  form  which  is  afterwards  reduced  by  the  action  of 
gravity  to  this  one,  so  on  this  ground  it  is  preferable  to  make 
the  cross-section  of  the  vessel  of  the  form  suggested.  Taking  all 
things  into  account,  a  curve  of  the  same  kind  that  has  been 
shown  to  be  the  best  for  the  water-lines,  appears  to  be  also  the 
best  for  the  cross-section ;  and  the  same  ordinates  which  answer 
for  the  water-lines  will  answer  for  the  cross-section,  only  in  the 
latter  case  the  ordinates  must  be  placed  closer  together.  If,  for 
example,  we  have  a  vessel  200  feet  long,  and  if  the  ordinates 
of  the  water-lines  be  5  feet  apart,  there  will  be  40  ordinates ; 
and  if  the  vessel  be  supposed  to  draw  20  feet  of  water,  the  same 
ordinates  placed  6  inches  apart  will  give  the  proper  form  of  the 
cross-section  below  the  load  water-line.  The  nearer  the  form 
of  the  cross-section  approaches  to  a  semicircle  the  less  friction 
there  will  be  in  the  vessel ;  and  the  proportions  of  the  cross- 
section  should  in  alt  cases,  where  practicable,  approach  to  the 
proportions  of  a  semicircle,  or  in  other  words  the  depth  below 


414 


STEAM    NAVIGATION. 


the  water  should  be  a  little  more  than  half  the  breadth  at  the 
water-line. 

The  ascending  water  will  move  more  and  more  rapidly  as  it 
comes  nearer  to  the  surface,  like  the  motion  of  a  falling  body  in- 
verted ;  and  its  momentum  will  carry  it  above  the  surface  to  a 
height  equal  to  that  which  would  generate  the  velocity.  This 
motion  of  the  water  above  the  surface  constitutes  the  second 
half  of  the  beat  of  the  pendulum  which  each  ascending  particle 
may  be  supposed  to  be — the  motion  of  the  particle  from  the  keel 
to  the  water-line  being  the  first  half  of  such  beat.  But  as,  after 
passing  the  surface  of  the  water,  the  particle  has  to  encounter 
more  of  the  power  of  gravity,  whereas  below  the  water  line  it  is 
floated  by  the  other  contiguous  particles,  it  will  follow  that  the 

Fig.  49. 


motion  of  the  particle  above  the  surface  will  be  smaller  in  the 
proportion  of  the  greater  retarding  force  it  there  has  to  encoun- 
ter. This  action  will  be  better  understood  by  a  reference  to 
fig.  48,  where  the  parallelogram  A  B  o  D  is  supposed  to  be  the 
side  of  a  ship,  w  L  is  the  surface  of  the  water  in  which  the  ship 
swims,  and  the  vertical  dotted  line  at  a  shows  the  position  of  the 
midship  frame.  If  we  suppose  a  particle  of  water  to  be  situated 
at  a;  a  little  below  the  water-level  at  the  bow,  then  as  the  vessel 
moves  onward  in  the  direction  of  the  arrow,  such  particle  will 
be  moved  upwards  faster  and  faster,  until  midway  between  the 
bow  and  the  midship  frame,  where  its  velocity  upwards  is  great- 
est, it  will  rise  above  the  surface  of  the  water  w  L,  and  its  own 
momentum  and  that  of  other  ascending  particles  will  carry  it 
upwards  until  it  reaches  the  position  of  the  midship  frame,  when 
it  will  begin  to  sink,  until  at  y  it  reaches  the  same  level  from 


MEASURE    OF   THE    HYDROSTATIC    RESISTANCE.          415 

which  it  rose.  The  surface  particles,  no  doubt,  which  terminate 
their  motion  at  y,  hegin  it  at  w  and  not  at  z,  and  to  this  circum- 
stance we  may  trace  the  origin  of  the  hydrostatic  resistance  of 
the  bow.  The  depression  at  y  will  be  as  great  below  the  mean 
water-level  w  L  as  the  elevation  at  a  is  above  it ;  and  if  the  sur- 
face of  the  water  at  the  stem  stood  at  x  instead  of  at  w,  the  fore- 
body  would  be  in  equilibrium,  seeing  that  the  depression  tx~w 
would  suck  the  vessel  forward  as  much,  or  nearly  so,  as  the  pro- 
tuberance from  t  to  a  would  impede  it.  As  the  hydrostatic 
pressure  from  a  to  s  pushes  the  vessel  forward  as  much  as  the 
depression  from  s  to  y  holds  it  back,  the  two  portions  of  the  after 
body  will  be  in  equilibrium ;  and  the  whole  moving  vessel  would 
be  in  equilibrium  if  the  surface  of  the  water  at  the  stem  stood  at 
x  instead  of  at  w.  As,  however,  the  water  stands  higher  at  the 
stem  than  at  the  stern,  there  will  be  a  hydrostatic  resistance  to 
be  encountered  which  is  equal  to  the  height  of  the  wave  midway 
between  a  and  w,  which  will  be  £a,  acting  against  the  breadth 
of  the  ship.  This  will  readily  be  understood  by  a  reference  to 
fig.  49^,  which  represents  a  horizontal  slice  of  a  floating  body  of 
the  height  of  the  wave  which  the  body  raises  in  passing  through 
the  water,  and  the  form  of  the  wave  is  represented  by  the  trian- 
gular figure  w  a  o,  which  is  delineated  on  the  plane  surface 
formed  by  cutting  away  one-quarter  of  the  model  so  as  to  clear 
the  problem  of  the  complication  involved  by  the  introduction  of 
the  curved  form  of  the  side.  A  transverse  ordinate  is  drawn  at 
&,  and  at  the  point  &  5,  where  this  ordinate  meets  the  side,  a  line 
is  drawn  parallel  to  the  axis,  intersecting  the  line  c  e.  From  the 
•point  of  intersection  a  vertical  line  5  is  raised,  on  which  is  set 
off  the  height  of  the  wave  at  &  &,  and  by  drawing  any  desired 
number  of  similar  lines  the  wave  w  a  c  will  be  set  off  on  the 
midship  section  in  the  form  ce  d,  which  figure  represents  the  hy- 
drostatic resistance  of  half  the  vessel.  The  area  of  the  figure 
c  e  d  is  manifestly  half  the  area  of  the  parallelogram  a  c  e  d ;  and 
as  there  is  a  similar  figure  on  the  other  side  of  the  vessel,  the 
total  area  representing  the  hydrostatic  resistance  will  be  equal  to 
half  the  height  of  the  wave  acting  against  the  breadth  of  the  ship. 
Supposing  that  no  disturbing  forces  were  in  existence  in  in- 


416 


STEAM   NAVIGATION. 


terfering  with  the  upward  and  downward  motion  of  the  water, 
a  particle  of  water  at  the  forefoot  B,  fig.  48,  would,  as  the  vessel 
moved  forward,  follow  the  curved  line  B  A;  and  if  on  rising 
above  the  lino  w  L  it  had  not  to  encounter  more  of  the  force 
of  gravity,  it  would  pursue  its  course  along  the  dotted  line  a  D. 

Fig.  49J. 


As,  however,  as  soon  as  the  particle  passes  ahove  w  L,  it  has  to 
encounter  nearly  the  whole  force  of  gravity,  its  momentum  will 
not  suffice  to  carry  it  up  far,  and  it  will  proceed  ahove  the  wa- 
ter level  only  to  some  such  point  as  a,  and  wiU  then  immedi- 
ately pass  downward  and  astern  in  the  track  of  the  curved  line 
a  o.  The  whole  of  the  ascending  and  descending  particles  will 
pursue  courses  nearly  parallel  to  these  tracks ;  and  such  lines 
might  be  drawn  mechanically  by  a  tracing  point  attached  to  a 


HOW   TO    REDUCE    LOSS    OF    MOMENTUM.  417 

pendulum  in  the  manner  already  described,  only  that  the  half 
of  the  beat  answering  to  the  motion  of  the  particle  above  the 
water-line,  would  be  reduced  in  length  by  the  ball  being  made 
in  this  part  of  its  motion  to  compress  a  spring  representing  the 
increased  power  of  gravity  to  which  the  particle  is  subjected 
during  this  part  of  its  course. 

Hitherto  we  have  discovered  no  source  of  loss  of  mechanical 
power  in  the  movement  of  the  water  by  a  vessel  passing  through 
it,  except  that  involved  by  the  necessity  of  overcoming  a  con- 
stant hydrostatic  resistance  in  consequence  of  the  difference  in 
the  level  of  the  water  at  the  bow  and  stern.  There  will,  how- 
Fig.  50. 


ever,  be  the  loss  of  the  momentum  left  in  the  undulating  mass 
of  water.  But  this  last  loss  will  be  diminished,  if  we  shift  the 
midship  frame  further  forward,  as  say  to  a,  fig.  50,  which  is  one- 
third  of  the  length  from  the  bow,  instead  of  half  the  length. 
For,  although  we  have  still  the  hydrostatic  resistance  equal  to 
half  the  height  of  a  above  w  L  multiplied  by  the  breadth  of  the 
vessel  to  encounter,  yet  if  the  after-body  of  the  vessel  be  prop- 
erly formed  with  diverging  sides,  the  undulating  mass  of  water 
will  have  surrendered  most  of  its  power  to  the  vessel  in  aid  of 
her  propulsion  before  it  leaves  the  stern  at  y.  If  we  snppose 
the  vessel  to  be  cut  off"  at  the  water  line,  we  shall  get  rid  of 
the  question  of  the  hydrostatic  resistance,  as  the  water  rising 
above  the  water-level  will  in  such  case  run  over  the  deck ;  but 
the  momentum  of  the  undulating  mass  will  remain,  and  the  ob- 
ject to  be  attained  is  so  to  form  the  stern  part  of  the  vessel  that 
the  upward  motion  of  the  water  above  the  water-line  at  the 
stern  shall  be  resisted,  whereby  the  mechanical  power  resident 
18* 


418  STEAM   NAVIGATION. 

in  the  heaving  water  will  be  communicated  to  the  vessel.  This 
is  done  at  present  practically  by  causing  the  stern  part  of  the 
vessel  to  spread  outwards  near  the  load  water-line,  so  that  the 
ascending  column  of  water  is  intercepted  by  it  and  gradually 
brought  to  rest. 

The  rise  of  water  at  the  bow,  it  will  be  observed,  increases 
not  merely  the  hydrostatic  pressure  against  which  the  vessel 
has  to  force  her  way,  but  also  the  opposing  area  against  which 
the  pressure  acts.  In  like  manner  the  deficient  height  of  water 
at  the  stern  diminishes  both  the  pressure  and  the  pressed  area. 
It  is  very  important,  therefore,  that  the  difference  of  level  at 
the  bow  and  stern  should  be  as  small  as  possible.  And  although 
we  have  supposed  that  the  height  of  the  wave  a,  fig.  50,  would 
only  be  the  same  if  we  shifted  forward  the  centre  frame,  it 
would  in  point  of  fact  be  higher  if  the  same  speed  of  vessel 
were  maintained.  On  this  ground,  therefore,  it  appears  prefera- 
ble to  maintain  the  midship  frame  near  the  position  shown  in 
fig.  48,  the  more  especially  as  the  forward  and  ascending  cur- 
rent due  to  the  friction  of  the  bottom  of  the  vessel  on  the  water 
has  a  tendency  to  bring  the  surface  of  the  water  relatively  with 
the  ship  into  the  condition  represented  by  the  waving-line 
x  t  a  s  y.  Before  entering  upon  the  consideration  of  the  friction 
of  the  bottom,  however,  it  may  be  stated  that  the  hydrostatic 
resistance  consequent  on  the  increased  elevation  beginning  at  w 
instead  of  at  x  is  not  all  loss.  For  while  the  height  of  the  wave 
increases  the  pressure  of  the  water  beneath,  it  also  helps  to  sep- 
arate the  water ;  and  if  the  vessel  be  made  without  any  straight 
part  between  the  fore  and  after-bodies,  a  portion  of  the  increased 
elevation  which  the  mean  water-line  w  L  receives  at  the  bow, 
will  be  retained  to  increase  the  elevation  of  the  water  at  the 
stern,  so  that  under  certain  conditions  nearly  the  whole  of  the 
power  expended  in  moving  the  water  would  be  theoretically  re- 
coverable. In  practice,  however,  such  a  result  is  never  reached ; 
and  however  perfect  the  arrangements  for  recovering  the  power 
may  be  made,  yet  a  certain  percentage  of  it  is  lost  at  every 
step ;  and  the  safest  indication  is  to  employ  such  a  form  of  vessel 
as  will  disturb  the  water  as  little  as  possible.  This  will  be  a 
body  of  the  form  which  I  have  indicated  with  a  considerable 


BEST    MODE    OF    SHAPING   VESSELS.  419 

proportion  of  length  to  breadth,  so  that  the  vessel  may  be  sharp 
at  the  ends.  A  length  of  7  times  the  breadth  is  found  to  be  a 
good  proportion  for  such  speeds  as  15  or  16  miles  an  hour.  But 
the  proportionate  length  that  is  advisable,  will  increase  with  the 
intended  speed. 

It  is  not  difficult  when  the  intended  speed  of  the  vessel  and 
also  its  length  and  breadth  are  determined,  to  find  what  the 
proper  form  of  the  vessel  will  be,  and  also  the  height  of  the 
wave  which  the  vessel  will  raise  at  the  midship  frame  by  her 
passage  through  the  water,  one-half  of  which  height  multiplied 
by  the  breadth  of  the  vessel  will  be  the  measure  of  the  hydro- 
static resistance.  For  as  each  particle  of  water  at  the  stern  has 
to  describe  the  motion  described  by  the  ball  of  a  pendulum 
which  makes  a  double  beat  during  the  time  that  the  vessel 
passes  through  her  own  length,  the  breadth  of  the  arc  will  an- 
swer to  half  the  breadth  of  the  vessel,  and  the  vertical  height  of 
the  arc  or  the  vertical  distance  fallen  by  the  ball  in  passing  from 
the  highest  to  the  lowest  part  of  the  arc,  will  be  the  height  of  the 
wave  raised  at  the  midship  frame — that  being  the  height  neces- 
sary to  give  the  velocity  of  motion,  with  which  the  particles  of 
water  must  be  moved  sideways  through  half  the  breadth  of  the 
vessel,  to  enable  the  vessel  to  pass  through  in  the  prescribed 
time.  If  we  suppose  the  ball  of  the  pendulum  to  be  replaced  by 
a  mass  of  liquid  moving  in  a  circular  arc,  the  motion  of  this 
liquid  will  be  the  same — except  in  so  far  as  it  is  affected  by 
friction — as  if  it  were  frozen  and  suspended  by  a  rod  of  the 
same  radius  as  the  arc ;  but  if  the  mass  of  liquid  be  large  so  as 
to  occupy  any  considerable  part  of  the  length  of  the  arc,  the 
motion  will  not  be  the  same  as  that  of  a  suspended  point,  as  the 
whole  of  the  particles  will  no  longer  rise  and  fall  through  the 
same  height,  while  all  of  them  will  have  still  to  be  moved  with 
the  same  velocity.  So  also  if  we  have  a  tube  open  and  turned 
up  at  both  ends,  and  if  we  pour  water  into  it  and  depress  the 
water  in  one  leg  so  as  to  disturb  the  equilibrium,  the  water 
when  released  will  vibrate  upward  and  downward  like  a  pendu- 
lum. Such  a  tube  is  represented  in  fig.  51,  where  E  A  B  n  is  the 
tube  which  is  filled  with  water  to  the  level  of  x.  If  the  level  in 
one  leg  be  depressed  from  o  to  G,  it  will  rise  in  the  other  leg 


420  STEAM    NAVIGATION. 

from  D  to  H  ;  and  if  the  depressing  force  be  now  withdrawn, 
the  water  will  fall  from  H  with  a  velocity  corresponding  to  its 
height  above  G,  and  will  be  carried  by  its  momentum  above  o 
to  E,  just  as  the  ball  of  a  pendulum  ascends  in 
£'}  '  its  arc  by  the  momentum  it  possesses — and  the 
water  will  continue  to  oscillate  np  and  down 
**  like  the  ball  of  a  pendulum,  until  it  is  finally 
D  brought  to  rest  by  friction.  If  the  tube  be  of 
equal  bore  throughout  and  be  bisected  in  o, 
then  as  the  accelerating  force  is  the  difference 
in  the  masses  of  the  two  unequal  columns  di- 
vided by  their  sum,  the  accelerating  force  will 
be  represented  by  E  G  divided  by  o  A  B  D,  or  what  is  the  same 
thing,  by  E  A  B  F  ;  or  it  will  be  proportional  to  the  half  of  this, 
or  to  E  o  divided  by  o  A  o.  The  time  of  the  oscillation  or  the 
time  in  which  the  surface  of  the  water  will  fall  from  the  highest 
to  the  lowest  point,  is  equal  to  that  in  which  a  pendulum  of  the 
length  o  A  o  makes  one  vibration.  Hence  the  time  in  which  the 
surface  will  pass  from  the  highest  point  to  the  lowest,  and  to 
the  highest  again,  will  be  that  in  which  a  pendulum  of  the 
length  o  A  o  will  make  two  vibrations,  or  it  will  be  that  in 
which  a  pendulum  of  four  times  that  length  makes  one  vibra- 
tion, or  a  centrifugal  pendulum  of  the  height  equal  to  o  A  o 
makes  one  revolution.  These  relations  equally  hold,  if  we  sup- 
pose the  same  kind  of  motion  which  exists  in  the  water  to  be 
produced  by  a  piston  at  o ;  and  the  side  of  the  ship  may  be 
supposed  to  be  such  a  piston,  and  if  properly  formed,  the  ship 
will  impart  sideways  to  the  water  precisely  the  same  kind  of 
motion  which  exists  in  the  case  here  illustrated. 

If  a  sheet  of  paper  be  drawn  vertically  behind  a  pendulum 
furnished  with  a  tracing  point,  then 
I'ig-  52.  if  the  pendulum  be  stationary,  the 

(2.)  tracing  point  will  draw  a  straight 

line  represented  by  the  dotted  line 
fig.  52.  But  if  the  pendulum  be 
put  into  motion,  then  the  tracer 
will  describe  the  waving  line  A  B  o  D 


SHARPNESS    SHOULD   VARY   WITH    SPEED. 


421 


Fig.  53. 


where  the  point  A  answers  to  the  stem  of  a  ship,  the  point 
B  to  the  midship  frame,  and  the  point  o  to  the  stern ;  and  the 
paper  will  pass  from  A  to  o  during  the  time  the  pendulum  makes 
two  oscillations.  Since  the  pendulum  has  to  make  two  oscilla- 
tions while  the  vessel  passes  through  a  distance  equal  to  her 
own  length,  the  combined  motions  of  the  tracer  and  pencil  will 
delineate  the  proper  form  for  the  side  of  the  vessel ;  and  if  made 
in  this  form  the  particles  of  water  will  have  the  same  motion  as 
the  ball  of  a  pendulum,  which  motion  enables  the  water  to  be 
moved  with  the  minimum  of  loss.  It  will  be  useful,  however, 
to  take  a  particular  case  to  show  in  what  manner  the  proper 
form  may  be  practically  determined. 

Suppose  A  c,  fig.  53,  to  represent  the  keel  of  a  vessel — which 
we  may  take  at  200  feet  long  and  40  feet  wide 
— and  which  is  intended  to  maintain  a  speed 
of  10  statute  miles  per  hour,  or  880  feet  per 
minute.  Now  as  the  vessel  has  to  pass 
through  her  length,  or  from  A  to  c,  during  the 
time  that  the  pendulum  p  makes  a  double 
beat,  or  to  pass  from  A  to  B,  which  is  100  feet, 
during  the  tune  the  pendulum  make  a  single 
beat,  there  will  be  880  divided  by  100,  or  8*8 
vibrations  of  the  pendulum  per  minute ;  and 
the  rod  of  the  pendulum  must  be  of  such " 
length  as  to  produce  that  number  of  vibra- 
tions. Now  to  determine  the  length  of  the 
rod  of  a  pendulum  which  shall  perform  any 
given  number  of  vibrations  per  minute,  we 
divide  the  constant  number  375-36  by  the 
number  of  vibrations  per  minute,  and  the 
square  of  the  quotient  is  the  length  in  inches. 
Hence  375-36  divided  by  8'8  =  42-6,  the  square 
of  which  is  1814*76  inches  or  151-23  feet,  and  apendulum  151-23 
feet  long  beating  in  an  arc  20  feet  long  with  the  paper  travelling 
at  a  speed  of  880  feet  per  minute,  will  describe  the  line  ABO, 
which  will  be  the  proper  water-line  for  the  side  of  a  ship  if  the 
cross-section  be  rectangular;  and  whatever  the  form  of  cross- 


422  STEAM    NAVIGATION. 

section  this  figure  will  equally  determine  the  proper  area  of 
cross-section  at  each  successive  frame.  If  instead  of  moving  at 
10  miles  an  hour,  the  vessel  has  only  to  move  at  the  rate  of  5 
miles  an  hour,  the  figure  described  will  be  that  represented  by 
D  a  E,  and  the  breadths  &  &  in  the  longer  figure  and  V  V  in  the 
shorter  are  the  same,  both  being  equal  to  half  the  breadth  at  a  a. 
The  rod  of  the  pendulum  p  p  passes  through  the  point  5,  and  the 
pendulum  vibrates  from  the  plane  of  the  keel  to  the  plane  of  the 
side,  so  that  the  chord  of  the  arc  in  which  the  vibration  is  per- 
formed is  equal  to  half  the  breadth  of  the  vessel,  while  the 
versed  sine  or  height  through  which  the  pendulum  falls  at  each 
beat,  will  be  equal  to  the  height  of  the  wave  at  the  midship 
frame.  To  find  the  versed  sine  of  the  arc,  we  divide  the  square 
of  half  the  chord  by  twice  the  length  of  the  pendulum.  The 
chord  being  20  feet  the  half  of  it  is  10  feet ;  and  the  pendulum 
being  151-23  feet  long  the  double  of  it  is  302'46  feet,  and  100 
divided  by  302-46  =  '33  feet  or  3'96  inches.  The  height  of  the 
wave  at  the  midship  frame,  in  a  vessel  formed  in  the  manner  in- 
dicated, will  accordingly  be  3'96  inches,  or  rather  this  would  be 
the  height  if  the  water  were  moved  without  friction,  so  that 
practically  the  height  will  be  somewhat  greater  than  is  here  in- 
dicated. 

If  we  increase  the  speed  of  the  vessel,  or  increase  the  breadth, 
the  hydrostatic  resistance  will  increase  very  rapidly.  Thus,  if 
the  speed  of  the  vessel  be  increased  to  20  miles  an  hour,  or  1,760 
feet  per  minute,  the  pendulum  will  require  to  make  1T6  beats 
per  minute,  and  its  length  will  be  375'36  divided  by  1T6  =  21'3, 
the  square  of  which  is  453'69  inches,  or  S'T'S  feet.  Now,  100 
divided  by  37'8  =  2'6  feet,  which  will  be  the  height  of  the  wave 
at  the  midship  frame  in  this  case,  and  the  hydrostatic  pressure 
will  be  the  half  of  this,  or  equivalent  to  1'3  feet  of  water  acting 
on  the  breadth  of  the  vessel.  In  like  manner,  successive  addi- 
tions to  the  breath  of  the  vessel  without  increasing  the  length 
add  rapidly  to  the  hydrostatic  resistance,  as  they  involve  the  ne- 
cessity of  the  oscillating  particles  ascending  higher  and  higher  in 
the  arc  to  enable  the  vessel  to  pass. 


FRICTION    OF   WATER.  423 

FRICTION  OF   WATER. 

It  remains  to  consider  the  friction  of  water  upon  the  bottom 
of  the  vessel,  and  this  is  by  much  the  most  important  part  of 
the  resistance  which  ships  have  to  encounter.  Beaufoy  made  a 
number  of  experiments  to  ascertain  the  amount  of  this  resistance 
by  drawing  a  long  and  a  short  plank  through  the  water :  and,  by 
taking  the  difference  of  their  resistances  and  the  difference  of 
their  surfaces,  he  concluded  that  the  friction  per  square  foot  of 
plank  was,  at  one  nautical  mile  per  hour,  '014  Ibs. ;  at  two 
nautical  miles  per  hour,  -0472  Ibs. ;  at  three,  -0948  Ibs. ;  four, 
•153  Ibs. ;  five,  -2264  Ibs. ;  six,  -3086  Ibs. ;  seven,  -4002  Ibs. ;  and 
eight,  '5008  Ibs.  At  two  nautical  miles  an  hour,  the  force  re- 
quired to  overcome  the  friction  was  found  to  vary  as  the  1*825 
power  of  the  velocity,  and  at  eight  nautical  miles  an  hour  as  the 
1'713  power.  Other  experimentalists  have  deduced  the  amount 
of  friction  from  the  diminished  discharge  of  water  flowing 
through  pipes.  If  there  were  no  friction  in  a  pipe,  the  velocity 
of  the  issuing  water  should  be  equal  to  the  ultimate  velocity  of 
a  body  falling  by  gravity  from  the  level  of  the  head  to  the  level 
of  the  orifice.*  But  as  the  velocity  is  found  by  the  diminished 
discharge  to  be  only  that  due  to  a  much  smaller  height,  the  dif- 
ference is  set  down  as  the  measure  of  the  power  consumed  by 
friction.  This  mode  of  estimating  the  friction  is  not  applicable 
to  the  determination  of  the  friction  of  a  ship ;  for,  in  the  first 
place,  the  discharge  is  a  measure  not  of  the  maximum,  but  of 
the  mean  velocity ;  and,  in  the  second  place,  there  is  every  reason 
to  believe  that  the  friction  per  square  foot  on  the  bottom  of  the 
ship  is  quite  different  near  the  bow  from  what  it  is  near  the 
stern.  As  the  water  adheres  to  the  bottom  there  will  be  a  film 
of  water  in  contact  with  the  ship,  which  will  be  gradually  pat 

*  There  is  sometimes  misconception  on  this  subject,  arising  from  a  neglect  of  the 
difference  between  the  ultimate  and  mean  velocities  of  a  falling  body.  Thus,  if 
water  flows  from  a  small  hole  in  the  side  of  a  cistern,  the  water  will  issue  with  tho 
ultimate  velocity  which  a  heavy  body  would  acquire  by  felling  from  the  level  of  the 
head  to  the  level  of  the  orifice,  which,  if  the  height  be  IS^j  feet,  will  be  82J  feet 
per  second.  The  mean  velocity  of  falling,  however,  is  only  16^  feet  per  second, 
BO  that  the  ultimate  velocity  is  twice  the  mean  velocity. 


424  STEAM    NAVIGATION. 

into  motion  by  the  friction ;  and  the  longer  the  vessel  is  the  less 
will  be  the  friction  upon  a  square  foot  of  surface  at  the  stern — 
seeing  that  such  square  foot  of  surface  has  not  to  encounter  sta- 
tionary water,  but  water  which  is  moving  with  a  certain  velocity  in 
the  direction  of  the  vessel.  The  film  of  water  moving  with  the  ves- 
sel will  become  thicker  and  thicker  as  it  passes  towards  the  stern, 
and  it  will  rise  towards  the  surface  by  reason  of  the  virtual  re- 
duction of  weight  consequent  upon  the  motion.  The  whole  of 
the  power,  therefore,  expended  in  friction  is  not  lost,  as  the 
power  expended  in  the  front  part  of  the  vessel  will  reduce  the 
friction  of  the  after  part ;  added  to  which,  the  rising  current 
which  the  friction  produces  may  be  made  to  aid  the  progress 
of  the  ship,  if  we  give  to  the  after-body  of  the  ship  such  a  con- 
figuration as  to  be  propelled  onward  by  this  rising  current. 
Finally,  when  the  screw  is  the  propelling  instrument,  the  slip 
of  the  screw  will  be  reduced,  and  may  even  in  some  cases  be 
rendered  negative,  by  the  circumstance  of  the  screw  working  in 
this  current ;  and  whatever  brings  this  current  to  rest  will  use 
up  the  power  in  it,  and  so  far  recover  the  power  which  has  been 
expended  in  overcoming  the  friction. 

In  my  investigations  respecting  the  physical  phenomena  of 
the  river  Indus  in  India,  I  observed  that  the  water  not  only  ran 
faster  in  the  middle  of  the  stream,  but  that  it  also  stood  higher 
in  the  middle,  so  that  a  transverse  section  of  the  river  would 
exhibit  the  surface  as  a  convex  line.  At  the  centre  of  the  river 
the  stream  is  very  rapid,  but  it  is  slow  at  the  sides,  so  that  boats 
ascending  the  river  keep  as  close  as  possible  to  either  bank ;  and 
in  some  parts  at  the  side  there  is  an  ascending  current  forming 
an  eddy.  I  further  observed,  that  not  merely  were  there  rapid 
and  considerable  changes  in  the  velocity,  which  I  imputed 
partly  to  the  agency  of  the  wind  in  deflecting  the  most  rapid 
part  of  the  current  to  the  one  side  or  the  other  of  the  river,  but 
there  were  also  diurnal  tides ;  or,  in  other  words,  the  stream  ran 
more  swiftly  in  the  afternoon  than  in  the  early  morning.  -This 
had  been  long  before  observed,  and  was  imputed  to  the  heat  of 
the  sun  melting  the  snows  in  the  mountains  more  during  the  day 
than  during  the  night.  But  although  such  an  effect  might  be 


INFLUENCE    OF   HEAT    ON   VELOCITY    OF   KIVEES.       425 

observable  in  a  single  feeder,  the  river  is  supplied  from  so  many 
sources  at  different  distances  tbat  such  intermittent  accessions 
would  equalise  one  another.  Moreover,  the  effect  of  the  sun  in 
the  daytime  in  swelling  the  volume  of  the  river,  if  acting  without 
any  equalising  influence,  could  only  produce  a  wave  like  a  tidal 
wave  in  the  river ;  and  the  increase  of  velocity  would  at  some 
points  take  place  at  night  and  at  some  in  the  morning,  whereas 
I  found  it  to  take  place  eceryichere  at  the  same  time.  I  finally 
came  to  the  conclusion  that  the  phenomenon  is  caused  by  the  in- 
fluence of  the  sun  in  heating  the  water  of  the  river,  and  thereby 
increasing  its  liquidity  and  its  velocity  throughout  the  whole 
length  of  the  river.  The  temperature  of  the  water  in  the  river 
is  commonly  about  94°  Fahr.,  but  as  the  river  is  wide  and  shal- 
low, it  is  rapidly  heated  and  cooled,  and  there  are  several  de- 
grees difference  between  the  temperature  of  the  day  and  the 
night.  In  the  early  morning  the  river  is  coldest,  and  at  that 
time  also — other  things  being  equal — its  velocity  is  least.  It 
may  hence  be  concluded  that  any  thing  which  gives  more  mo- 
bility to  the  particles  of  the  water  in  which  a  vessel  floats  will 
diminish  the  friction  of  the  bottom ;  and  this  end  seems  likely  to 
be  attained  by  the  injection  of  air  into  the  water  at  the  stem 
and  forefoot  or  front  part  of  the  keel. 

It  is  not  difficult  to  understand  how  it  comes  that  the  water 
hi  a  river  should  stand  higher  at  the  middle  than  at  the  sides,  as 
shown  in  fig.  54.    If  we  hang  a  weight  upon  a  spring  balance 
we  shall  find  the  amount  of  the  weight  to 
be  indicated  on  the  scale  or  index ;  and  Fig.  54 

this  weight  will  continue  to  be  shown  so 
long  as  we  hold  the  spring  balance  sta- 
tionary. But  if  we  allow  it  to  move  tow- 
ards the  earth  with  the  velocity  which 
a  heavy  body  would  acquire  in  falling  by  gravity,  the  index  of 
the  spring  will  show  no  tension  at  all — proving  that  with  this 
amount  of  downward  motion  the  body  imparts  no  weight.  If 
the  spring  is  moved  downward  slower  than  a  body  falls  by  grav- 
ity, the  spring  will  show  that  it  is  sustaining  some  weight ;  but 
at  any  velocity  downward  there  will  be  a  diminution  in  the 


426 


STEAM    NAVIGATION. 


Fig.  55. 


weight  of  the  body  answerable  to  that  velocity.  In  two  columns 
of  water,  therefore,  moving  at  different  velocities,  the  slower 
will  exert  most  hydrostatic  pressure  on  the  pipe  or  channel  con- 
taining it ;  and  where  two  such  columns  are  connected  together 
sideways,  as  in  a  river,  the  faster  must  rise  to  a  greater  height  to 
be  in  hydrostatic  equilibrium  sideways  with  the  slower.  The 
surface  of  the  water  consequently  becomes  convex,  as  shown  at 
M  in  fig.  54,  where  H  is  the  water  and  A  B  o  D  the  bed. 

It  will  be  seen  from  these  observations  that  there  is  a  hy- 
draulic as  well  as  a  hydrostatic  head  of  water ;  and  the  hydraulic 
head  is  equal  to  the  hydrostatic  head, 
diminished  by  the  height  due  to  the 
velocity  with  which  the  water  flows. 
This  law  is  further  illustrated  by  fig. 
55,  which  represents  a  bulging  vessel 
in  which  the  water  is  maintained  at  a 
uniform  height  by  water  flowing  into 
it  at  the  top,  while  it  runs  out  at  E  at 
the  bottom.  The  velocity  with  which 
the  water  flows  downward  from  A  to 
E,  varies  with  the  amount  of  enlarge- 
ment or  contraction  of  the  vessel; 
and  the  height  of  water  which  will 
be  supported  in  the  small  pipes  J,  c 
and  <Z,  varies  as  the  velocity  of  the  water  at  their  several  points 
of  insertion.  Thus,  the  area  at  B,  being  greater  than  the  area 
at  A,  the  velocity  will  be  less,  and  consequently  the  water  will 
stand  in  the  small  pipe  5  at  a  point  higher  than  the  surface  of  A. 
The  area  at  D  being  less  than  the  area  at  A,  the  velocity  will  be 
greater ;  and  the  height  of  the  water  in  the  small  tube  d  will  not 
come  up  to  the  level  of  A.  At  o,  the  velocity  of  the  water  being 
very  great,  not  only  no  height  of  column  will  be  supported  in  the 
tube  c  there  inserted,  but  the  water  will  be  sucked  up  through 
the  inverted  tube  c,  out  of  the  small  cistern  F  ;  and  if  there  be 
no  cistern  air  will  be  drawn  through  the  tube.  So  also  in  fig. 
56,  if  a  pipe  be  led  out  at  the  bottom  of  a  cistern  of  water,  a 


FRICTION    OF   THE    BOTTOM    OF    VESSELS. 


427 


Fig.  56.  hole  bored  in  any  part  of  the  pipe  will  draw  air  and 
not  leak  water,  so  long  as  the  water  is  running  out 
of  the  bottom  of  the  pipe. 

It  follows,  from  these  considerations,  that  the 
stratum  of  water  put  into  motion  by  the  friction  of 
the  vessel  will  rise  to  a  higher  level  than  the  sur- 
rounding water,  which  is  at  rest ;  and  advantage 
should  be  taken  of  this  ascending  current  to  aid  in 
propelling  the  vessel,  by  spreading  out  the  stern  part 
so  as  to  intercept  and  derive  motion  from  the  rising 
water.  This  is,  to  some  extent,  done  in  common  ves- 
sels by  the  greater  breadth  which  is  given  to  the 
stern  part  near  the  water  level;  and  although  no 
very  tangible  reason  is  commonly  adduced  for  the  practice  be- 
yond that  of  affording  greater  accommodation  for  the  cabins,  the 
method  of  expanding  the  breadth  at  the  stern  is  also  useful  in 
utilising  the  ascending  current.  The  manner  in  which  the  ship 
acts  upon  the  water  in  urging  it  into  motion  by  friction  is  not 
known.  But  it  is  known  that  the  vessel  carries  a  film  of  water 
with  it  in  the  same  manner  as  the  belt-pump ;  and  it  is  known 
that  the  particles  of  water  nearest  the  vessel  move  with  a  velocity 
nearly  the  same  as  that  of  the  vessel,  and  that  the  motion  of  each 
particle  diminishes  in  amount  the  further  it  is  from  the  vessel, 
until  those  particles  are  reached  which  are  wholly  at  rest.  The 
moving  film  may  consequently  be  regarded  as  a  roller  interposed 
between  the  bottom  of  the  vessel  and  the  water ;  and  such  a 
roller  would  enable  the  vessel  to  move  forward  with  twice  the 
speed  that  the  roller  itself  moves  at.  But  before  this  roller  can 
be  set  into  motion,  there  will  be  a  good  deal  of  slip  or  pure  fric- 
tion, just  as  there  is  in  the  driving-wheel  of  a  locomotive  in 
starting  the  train.  It  is  not  known  what  length  of  vessel  will 
sulfice  to  move  the  film  of  water  with  the  maximum  velocity  it 
can  attain  with  any  given  speed  of  the  ship ;  nor  is  it  known 
what  the  maximum  speed  of  the  film  is  with  any  given  velocity 
of  the  ship.  The  speed  will  always  be  less  than  the  speed  of  the 
ship,  but  how  much  less  is  not  known ;  and  this  speed,  when 
once  attained,  will  not  be  increased,  as  when  it  is  reached  the 


428  STEAM   NAVIGATION. 

power  communicated  by  the  friction  of  the  bottom  will  be 
balanced  by  the  power  consumed  in  maintaining  the  motion 
among  the  internal  particles.  Up  to  a  certain  point,  therefore, 
the  friction  upon  a  square  foot  of  the  ship's  bottom  will  diminish 
with  the  distance  from  the  stem ;  and  the  thickness  of  the  moving 
film  will  also  increase  with  that  distance.  But  when  that  point 
of  the  length  has  been  reached,  the  friction  per  square  foot  will 
become  uniform,  and  there  will  be  no  further  increase  in  the 
thickness  of  the  film. 

Instead,  however,  of  supposing  the  film  interposed  between 
the  stationary  water  and  the  moving  bottom  to  be  a  single 
roller,  it  will  be  a  nearer  approximation  to  the  truth  if  we  sup- 
pose it  to  be  composed  of  an  infinite  number  of  rollers,  a  a  a  a 
in  fig.  57,  where  we  may  suppose  s  s  to  be  the  ship,  while  the 
line  extending  from  roller  to  roller  represents  the 
amount  of  motion  which  the  water  receives  from 
each  successive  length  of  the  ship,  and  which  dimin- 
ishes  as  we  recede  from  the  stem  until  we  reach  the 
point  A  B,  where  the  pure  friction  of  the  bottom 
upon  the  particles  balances  the  power  consumed  in 
maintaining  the  internal  motion  of  the  water,  and 
which  power  is  ultimately  transformed  into  heat. 
The  whole  power  concerned  in  propelling  the  ves- 
sel  is  consumed  either  in  moving  the  water  or  in 
heating  it.    The  greater  part  of  the  power  expended 
in  moving  the  water  aside  at  the  bow,  is  recovered 
~\    v     i      by  the  closing  of  the  water  at  the  stern ;  and  most 
of  that  expended  in  friction  in  producing  a  rising 
current  is  recoverable  by  giving  a  proper  configuration  to  the 
stern.     Of  the  heat  generated,  the  whole  is  not  lost,  as  it  will 
give  greater  mobility  to  the  particles  of  the  water,  which  will 
also  be  given  by  heating  the  bottom,  as  has  been  done  in  some 
steam-vessels,  by  converting  the  bottom  into  a  refrigerating  sur- 
face for  condensing  the  steam ;  and  by  which  arrangement  the 
bottom  itself  has  been  heated  to  some  extent.     On  the  whole, 
however,  that  arrangement  will  be  the  most  advantageous  for 
reducing  resistance  by  which  the  least  motion  is  given  to  the 


SPEED    OP   VESSELS   OF   A   GIVEN   POWER.  429 

water,  and  the  least  heat  generated  in  it ;  and  the  smoothness 
of  the  rubbing  surface  will  somewhat  affect  that  question.  In 
pipes,  it  has  been  found  that  there  is  no  increase  of  friction  from 
increase  of  pressure.  But  it  must  not  be  therefore  inferred  that 
in  vessels  the  friction  per  square  foot  is  precisely  the  same 
at  every  point  in  the  depth,  any  more  than  at  every  point  of 
the  length  ;  for  the  moving  water  has  to  escape  to  the  surface, 
and  the  difficulty  of  the  escape  will  be  the  greater  the  further 
the  surface  is  off.  If  we  knew  the  ratio  in  which  the  resistance 
of  a  vessel  increased  with  the  length  and  with  the  depth,  we 
should  be  able  to  tell  what  form  the  vessel  should  have,  in  order 
to  offer  the  least  resistance.  But  it  is  quite  certain  that  the  re- 
sistance per  square  foot  of  the  bottom  does  diminish  with  the 
length  in  some  proportion  or  other ;  and  as  the  resistance  also 
diminishes  as  the  wetted  perimeter,  and  as  relatively  with  the 
sectional  area,  the  wetted  perimeter  of  large  vessels  is  less  than 
that  of  small,  it  is  easy  to  understand  how  it  comes  that  large 
vessels  are  swifter  than  small  with  the  same  proportion  of  pro- 
pelling power.  If  we  double  the  breadth  and  immersed  depth 
of  a  vessel,  we  double  the  length  of  its  perimeter.  But  we  in- 
crease its  sectional  area  fourfold ;  and  as  with  any  given  length, 
and  with  equally  fine  ends,  the  wetted  perimeter  is  the  measure 
of  the  resistance,  it  follows  that  the  large  vessel  will  require  less 
power  per  ton  or  per  square  foot  of  immersed  section  to  main- 
tain any  given  speed. 

SPEED  OF  STEAM  VESSELS  OF  A  GIVEN  POWER. 

There  were  no  accepted  rules  for  ascertaining  the  speed  that 
a  steam  vessel  of  a  given  type  would  probably  obtain  with  en- 
gines of  a  given  power,  until  the  appearance  of  the  first  edition 
of  my  Catechism  of  the  Steam-Engine,  when  I  published  the  rule 
which  had  long  been  employed  by  Messrs.  Boulton  and  Watt 
for  determining  this  point.  This  rule  was  founded  on  a  long- 
continued  series  of  experiments  on  steam  vessels  of  different 
types ;  and  for  similar  kinds  of  vessels  the  results  it  gives  have 
been  found  very  nearly  to  accord  with  those  subsequently  ob- 


430  STEAM    NAVIGATION. 

tained  by  experiment.  This  rule,  which  proceeds  on  the  suppo- 
sition that  the  engine  power  required  for  the  propulsion  of  a 
vessel  varies  as  the  area  of  the  immersed  midship-section,  and  as 
the  cube  of  the  speed,  has  been  already  referred  to  in  page  77 
as  an  example  of  the  application  of  equations,  and  in  algebraical 
language  it  is  as  follows : — 

If  s  be  the  speed  of  the  vessel  in  knots  per  hour,  A  the  area 
of  the  immersed  midship  section  in  square  feet,  o  a  numerical 
coefficient,  varying  with  the  form  of  vessel  and  to  be  fixed  by 
experiment,  and  p  the  indicated  horse-power :  then 

83A,  83A  ,  PC 

p  =  — -        o  =  —        and        s  =  Jy  — 
OP  v  A 

In  words  these  rules  are  as  follows : — 


TO   DETERMINE   THE   POWEE  NECESSARY  TO  REALISE  A  GIVEN  SPEED 
IN  A  STEAM  VESSEL  BY  BOTJLTON  AND  WATT'S  EULE. 

RULE. — Multiply  the  cube  of  the  given  speed  ly  the  area  in 
square  feet  of  that  part  of  the  midship  section  of  the  vessel 
lying  lelow  the  water-line,  and  divide  the  product  ~by  a  cer- 
tain coefficient  of  which  there  is  a  different  one  for  each 
particular  type  of  vessel.  The  quotient  is  the  indicated 
power  in  horses  that  will  le  required  to  give  the  intended 
speed. 

Example. — The  steamer  '  Fairy,'  with  an  immersed  sectional 
area  of  71-J-  square  feet,  and  a  coefficient  of  465,  attained  on 
trial  a  speed  of  13'3  knots  per  hour.  What  indicated  power 
must  have  been  exerted  to  attain  this  speed  ? 

Here  the  cube  of  13'3  is  2352-637,  which  multiplied  by  71'5 
=  168210'9,  and  this  divided  by  465  is  equal  to  363  horse-power, 
which  was  the  power  actually  exerted  in  this  case. 

In  the  first  edition  of  my  Catechism  of  the  Steam-Engine 
the  coefficients  of  a  number  of  steam-vessels  were  given,  which 
had  been  ascertained  experimentally  by  Boulton  and  "Watt ;  and 
in  the  first  edition  of  my  Treatise  on  the  Screw  Propeller,  pub- 
.lished  in  1852,  I  recapitulated  a  number  of  the  coefficients  of 


RULES    FOR   FINDING    SPEED    OF    VESSELS.  431 

the  screw  steamers  of  the  navy,  which  had  then  heen  recently 
ascertained  by  the  steam  department  of  the  navy,  as  also  the  co- 
efficients obtained  by  multiplying  the  cube  of  the  speed — not  by 
the  area  of  the  midship  section,  but  by  the  cube  root  of  the 
square  of  the  displacement — and  dividing  by  the  indicated  power. 
The  displacement  of  the  '  Fairy  '  at  the  trial,  at  which  the  speed 
was  13'3  knots,  was  168  tons.  Now  the  square  of  168  is  28224, 
the  cube  root  of  which  is  30*45  nearly,  and  this  multiplied  by  the 
cube  of  the  speed  2352'637  and  divided  by  the  indicated  power, 
363  horses,  gives  197  as  the  coefficient  proper  to  be  employed 
when  this  measure  of  the  resistance  is  adopted.  Neither  the 
immersed  section,  however,  nor  the  displacement,  is  the  proper 
measure  of  the  resistance  in  steam  vessels ;  and  I  pointed  this 
out  in  the  first  edition  of  my  Treatise  on  the  Screw  Propeller,  in 
1852,  and  suggested  the  wetted  perimeter  as  a  preferable  meas- 
ure of  the  resistance ;  the  perimeter  being  a  measure  of  the 
friction ;  and  nearly  the  whole  of  the  resistance  of  well-formed 
ships  being  produced  by  friction.  Under  this  view  the  velocity 
of  ships  with  any  given  perimeter  and  propelling  power  would 
fall  to  be  considered  in  much  the  same  way  as  the  velocity  of  the 
water  flowing  hi  rivers  or  canals,  and  in  which  the  speed  with 
any  given  declivity  of  the  bed  varies  as  the  hydraulic  mean 
depth,  or  in  other  words  as  the  sectional  area  of  the  stream  di- 
vided by  the  wetted  perimeter.  In  such  a  comparison  the  en- 
gine power  of  the  ship  answers  to  the  gravitation  of  the  stream 
down  the  inclined  plane  of  the  bed,  while  the  area  of  the  trans- 
verse section  of  the  ship  beneath  the  water-line  divided  by  the 
wetted  perimeter  constitutes  the  hydraulic  mean  depth  of  the 
ship.  This  measure  of  the  resistance,  however,  though  accurate 
enough  for  short  vessels,  is  not  applicable  to  long  vessels  without 
some  allowance  being  made  for  the  inferior  resistance  of  long 
vessels  of  the  same  sharpness  at  the  ends,  in  consequence  of  the 
proportion  of  power  which  long  vessels  recover,  especially  if 
propelled  by  the  screw  or  any  other  propeller  situated  at  the 
stern. 


432  STEAM  NAVIGATION. 

TO    DETERMINE   BOULTON  AND  WATT'S  COEFFICIENT  FOE  ANY  GIVEN 
VESSEL  OF  WHICH  THE  PERFORMANCE  IS  KNOWN. 

RULE. — Multiply  the  cube  of  the  speed  in  knots  per  hour  l>y  the 
area  in  square  feet  of  the  immersed  transverse  section  of  the 
vessel,  and  divide  the  product  ~by  the  indicated  horse-power. 
The  quotient  will  5e  the  coefficient  of  that  particular  type 
of  vessel. 

Example. — The  steamer  Fairy,  with  an  area  of  immersed 
section  of  11^  square  feet,  and  363  indicated  horse-power,  at- 
tained a  speed  of  13'3  knots  an  hour.  "What  is  the  coefficient 
of  that  vessel? 

Here  13'3  cubed  =  2352-637,  which  multiplied  by  71 '5  and 
divided  by  363  horse-power  =  465,  which  is  the  coefficient  of 
this  vessel  according  to  Boulton  and  Watt's  rule.  A  good  num- 
ber of  coefficients  for  different  vessels  is  given  at  page  77. 

TO  DETERMINE  WHAT  SPEED  WILL  BE  ATTAINED  BY  A  STEAM  VESSEL 
OF  A  GIVEN  TYPE  WITH  A  GIVEN  AMOUNT  OF  ENGINE  POWER,  BY 
BOULTON  AND  WATT'S  EULE. 

RULE. — Multiply  the  indicated  horse-power  "by  the  coefficient 
proper  for  that  particular  type  of  vessel,  and  divide  the 
product  J)y  the  area  of  the  immersed  transverse  section  in 
square  feet.  Extract  the  cube  root  of  the  quotient,  which, 
will  l)e  the  speed  that  will  lie  obtained  in  knots  per  hour. 

Example. — What  speed  will  be  obtained  in  a  steamer  of 
which  the  coefficient  is  465,  and  which  has  an  immersed  section 
of  71 J  square  feet,  and  is  propelled  by  engines  exerting  363 
horse-power. 

Here  363  x  465=168795,  which  divided  by  71 '5=2360.  The 
logarithm  of  this  is  3-372912,  which  divided  by  3=1-124304, 
the  natural  number  answering  to  which  is  1331.  Now  the  in- 
dex of  the  divided  logarithm  being  1,  there  will  be  two  integers 
in  the  natural  number  answering  to  it,  which  will  consequently 
be  13-31,  and  this  will  be  the  speed  of  the  vessel  in  knots  per 
hour. 


MEAN    VELOCITY   OF   WATEB    IN    CANALS,    ETC.         433 

The  coefficient  of  a  steamer  sometimes  varies  with  the  speed 
with  which  the  vessel  is  propelled.  If  the  vessel  is  properly 
formed  for  the  speed  at  which  she  is  driven,  then  her  coefficient 
will  not  become  greater  at  a  lower  speed ;  and  if  it  becomes 
greater,  the  circumstance  shows  that  the  vessel  is  too  blunt. 
When  the  '  Fairy'  was  sunk  to  a  draught  of  5  feet  10  inches,  her 
speed  was  reduced  to  11-89  knots,  and  her  coefficient  was  re- 
duced from  465  to  429,  showing  that  she  worked  more  advan- 
tageously at  the  higher  speed  and  lighter  draught.  The  '  War- 
rior,' which  when  exerting  5,469  horse-power  attained  a  speed 
of  14.356  knots  with  a  coefficient  of  659,  attained  when  exerting 
2,867  horse-power  a  speed  of  12-174  knots  with  a  coefficient  of 
767 ;  and  when  exerting  1,988  horse-power  a  speed  of  11-040 
knots  with  a  coefficient  of  825.  This  shows  that  the  'Warrior' 
is  too  blunt  a  vessel  for  a  high  rate  of  speed. 

It  will  be  satisfactory  to  ascertain  the  comparative  eligibility 
of  the  forms  of  the  'Fairy'  and  the  "Warrior,'  which  we  may 
easily  do  by  comparing  the  speed  attained  by  each,  with  the 
speed  which  would  be  attained  by  an  equal  weight  of  water 
running  in  a  river  or  canal,  and  impelled  by  an  equal  motive 
force.  The  rule  for  determining  the  speed  of  water  flowing  in 
rivers  or  canals  of  any  given  declivity  is  as  follows  : — 

TO  DETERMINE  THE  MEAN  VELOCITY  WITH  WHICH  WATEE  WILL 
PLOW  THROUGH  CANALS,  AETEKIAL  DRAINS,  OE  PIPES,  SUNNING 
PARTLY  OE  WHOLLY  FILLED. 

RULE. — Multiply  the  hydraulic  mean  depth  in  feet  "by  twice  the 
fall  in  feet  per  mile.    Extract  the  square  root  of  the  product, 
which  i*  the  mean  velocity  of  the  stream  in  feet  per  minute. 
Now  the  'Fairy,'  when  realizing  a  speed  of  13'3  knots  per 
hour  with  363  horse-power,  had  a  draught  of  water  of  4-8  feet ; 
a  sectional  area  of  71'5  feet;  a  wetted  perimeter  of  24'7  feet, 
and  a  displacement  of  168  tons.    The  hydraulic  mean  depth  be- 
ing the  sectional  area  in  square  feet,  divided  by  the  length  of 
the  wetted  perimeter  in  feet,  the  hydraulic  mean  depth  will  in 
this  case  be  71  -5,  divided  by  24-7=2-9. 

The  engine  made  51-6  revolutions  per  minute,  and  the  screw 
19 


434  STEAM   NAVIGATION 

258  revolutions  per  minute,  being  five  times  the  number  of  rev- 
olutions of  the  engines.  The  stroke  is  3  feet,  and  the  pitch  of 
the  screw  8  feet. 

Now  a  horse-power  being  33,000  Ibs.,  raised  1  foot  per  min- 
ute, and  as  there  were  363  horse-power  exerted,  the  total  effort 
of  the  engines  will  be  363  times  33,000,  or  11,979,000  Ibs.,  raised 
through  1  foot  each  minute.  But  the  engine  makes  51 '6  revolu- 
tione  each  minute,  and  the  length  of  the  double  stroke  is  6  feet, 
so  that  the  piston  moves  through  309 '6  feet  per  minute;  and 
the  power  being  the  product  of  the  velocity  and  the  pressure, 
the  power  11,979,000  Ibs.  divided  by  the  velocity  of  the  piston, 
309'6  feet  per  minute,  will  give  the  mean  pressure  urging  the 
pistons,  which  will  be  38,691  Ibs.  But  the  speed  of  the  screw- 
shaft  being  five  times  greater  than  that  of  the  engine-shaft,  the 
pressure  urging  it  into  revolution  must,  in  order  that  there  may 
be  an  equality  of  power  in  each,  be  five  times  less ;  or  it  will  be 
7,538  Ibs.  moving  through  6  feet  at  each  revolution.  Then  the 
pitch  of  the  screw  being  8  feet,  the  thrust  of  the  screw  will  be 
less  than  7,538,  in  the  proportion  in  which  6  is  less  than  8,  or  it 
will  be  5,653,  supposing  that  there  is  no  loss  of  power  by  slip 
and  friction.  It  is  found  on  an  average  in  practice,  that  about 
one-third  of  the  power  is  lost  in  slip  and  friction ;  and  the  actual 
thrust  of  the  screw-shaft  will  be  about  one-third  less  than  the 
theoretical  thrust,  or  in  this  case  it  will  be  3,769  Ibs.  or  1*68 
ton.  Now,  in  order  that  168  tons  of  water  may  gravitate  down 
an  inclined  channel  with  a  weight  of  1*68  ton,  the  declivity  of 
the  channel  must  be  1  in  100.  In  1  mile,  therefore,  it  will  be 
52-80  feet.  A  cubic  foot  of  salt-water  weighs  64  Ibs.,  so  that 
there  are  35  cubic  feet  in  the  ton,  and  in  168  tons  there  are 
5,880  cubic  feet.  Dividing  this  by  the  sectional  area  71 '5  feet, 
we  get  a  block  of  water  82'2  feet  long,  and  with  a  cross-section 
of  71 '5  feet,  weighing  168  tons;  and  the  wetted  perimeter  being 
24*7  feet,  and  the  length  82'2  feet,  we  get  a  rubbing  area  of 
2020*34  feet;  and  as  the  friction  on  this  surface  balances  the 
weight  of  3,769  Ibs.,  there  will  be  a  friction  of  1*8  Ib.  on  each 
square  foot.  If  this  block  of  water  be  supposed  to  be  let  down 
a  channel  falling  1  in  100,  its  velocity  will  go  on  increasing  until 


SHIPS    COMPARED    WITH   RIVERS.  435 

the  friction  balances  the  gravity,  which,  according  to  the  rule 
given  above,  "will  be  when  the  water  attains  a  speed  of  11  miles 
an  hour,  from  whence  we  conclude  that  the  sum  of  the  resist- 
ances of  a  well-formed  skip  are  less  than  the  friction  alone  of  an 
equal  weight  of  water  of  the  same  hydraulic  depth,  moved  in  a 
pipe  or  canal  by  an  equal  impelling  force.  If  instead  of  taking 
the  declivity  in  2  miles,  as  the  rule  prescribes,  to  ascertain  the 
velocity  of  the  water,  we  take  the  declivity  in  twice  2,  or  4 
miles,  we  shall  arrive  at  a  pretty  exact  expression  of  the  speed 
of  the  vessel  in  this  particular  case.  Taking  the  knot  at  6,101 
feet,  13'3  knots  will  be  equal  to  15'3  statute  miles,  and  the  de- 
clivity in  1  mile  being  52-8  feet,  the  declivity  in  4  miles  will  be 
211'2  feet.  Multiplying  this  by  2'9,  the  hydraulic  mean  depth, 
we  get  612'48,  the  square  root  of  which  is  24*7,  which  multi- 
plied by  55,  gives  the  speed  of  the  water  in  feet  per  minute = 
1,358-5.  This,  multiplied  by  60,  gives  81,510  feet  as  the  speed 
per  hour,  and  this  divided  by  5,280,  the  number  of  feet  in  a 
statute  mile,  gives  15 '4  as  the  speed  in  statute  miles  per  hour. 

The  'Warrior,'  with  a  displacement  of  8,852  tons,  a  draught 
of  water  of  25  J  feet,  an  immersed  midship  section  of  1,219  square 
feet,  and  5,469  horse-power,  attained  a  speed  of  14*356  knots,  or 
16'6  statute  miles.  The  number  of  strokes  per  minute  was  34J, 
and  the  length  of  the  double  stroke  8  feet,  while  the  pitch  of  the 
screw  was  30  feet.  The  wetted  perimeter  is  88  feet,  which 
makes  the  mean  hydraulic  depth  13 '8  feet.  The  power  being 
5,469  horses,  33,000  times  this,  or  180,477,000  Ibs.,  will  be  lifted 
1  foot  high  per  minute.  But  as  the  piston  travels  54'25  times  8 
feet,  or  434  feet  each  minute,  the  load  upon  the  pistons  will  be 
415,845  Ibs.  The  pitch  of  the  screw,  however,  being  30  feet, 
while  the  length  of  a  double  stroke  is  8  feet,  the  theoretical 
thrust  of  the  screw  will  be  reduced  in  the  proportion  in  which 
30  exceeds  8,  or  it  will  be  110,892  Ibs.  If  from  this  we  take 
one-third,  on  account  of  losses  from  slip  and  friction,  we  get 
73,928  Ibs.,  or  33  tons,  as  the  actual  thrust  of  the  screw. 

Now  8,852,  which  is  the  displacement  in  tons,  divided  by 
83  tons,  which  is  the  motive  force  in  tons,  gives  268,  or,  in  other 
words,  the  declivity  of  the  channel  must  be  1  in  268,  in  order 


46b  STEAM    NAVIGATION. 

that  8,852  tons  may  press  down  the  inclined  plane  with  a  force 
of  33  tons.  This  is  a  declivity  of  very  nearly  20  feet  in  the 
mile,  or  40  feet  in  two  miles,  or  80  feet  in  twice  two  miles. 
The  mean  hydraulic  depth  being  13'8  feet,  80  times  this  is  1,104, 
the  square  root  of  which  is  33'2,  which  multiplied  by  55  =  1,826 
feet  per  minute,  or  multiplying  by  60=109,500  feet  per  hour. 
Dividing  by  5,280,  we  get  the  speed  of  20  miles  per  hour,  which 
ought  to  be  the  speed  of  the  '  Warrior '  if  her  form  were  as 
eligible  as  that  of  the  'Fairy.'  The  speed  falls  3'4  miles  an  hour 
short  of  this,  which  defect  must  be  mainly  imputed  to  the  de- 
ficient sharpness  of  the  ends  for  such  a  speed  and  draught,  and 
the  increased  resistance  consequent  on  the  greater  depth. 

In  a  paper  by  Mr.  Phipps,  on  the  '  Eesistances  of  Bodies  pass- 
ing through  "Water,'  read  before  the  Institution  of  Civil  En- 
gineers in  1864,  it  was  stated  that  these  resistances  comprised 
the  Plus  Resistance,  or  that  concerned  in  moving  out  of  the  way 
the  fluid  in  advance  of  the  body ;  the  Minus  Eesistance,  or  the 
diminution  of  the  statical  pressure  behind  any  body  when  put 
into  a  state  of  motion  in  a  fluid ;  and  the  Frictional  Resistance 
of  the  surface  of  the  body  in  contact  with  the  water. 

The  Plus  Resistance  of  a  plane  surface  one  foot  area,  moving 
at  right  angles  to  itself  in  sea  water,  was  considered  to  be 

64'2  x  02 

It  = -,  and  the  Minus  Resistance  was  one  half  the  Plus 

2? 
Resistance. 

For  planes  moving  in  directions  not  at  right  angles  to  them- 
selves, the  theoretical  resistances  were,  for  the  Plus  Pressure — 

a         ,     „      #64-2fl2 

8  =  -,  and   B  =  —    — , 

r8'  2g    ' 

the  Minus  Pressure  being  one-half  the  above ;  where  JK  was  the 
resistance  of  the  inclined  plane ;  «,  the  area  of  the  projection 
of  the  inclined  plane  upon  a  plane  at  right  angles  to  the  direc- 
tion of  motion ;  r,  the  ratio  of  the  areas  of  the  projected  and 
the  inclined  planes ;  and  $,  the  area  of  a  square-acting  plane 
of  equivalent  resistance  with  the  inclined  plane. 

But,  besides  these  theoretical  resistances,  the  experiments  of 
Beaufoy  showed,  that  when  the  inclined  planes  were  of  moderate 


RECENT    COMPUTATIONS    OF   RESISTANCE.  437 

length  only,  the  Plus  Resistance  was  considerably  in  excess  of 
the  above ;  so  that  when  the  slant  lengths  of  the  planes  were  to 
their  bases  in  the  proportion  of 

2  to  1,  3  to  1,  4  to  1,  and  6  to  1, 

the  actual  resistances  exceeded  the  theoretical,  as 

1-1  to  1,  1-98  to  1,  3-24  to  1,  and  6'95  to  1. 

Mr.  Phipps  proposed  a  method  of  approximating  to  these  ad- 
ditional resistances,  by  adding  the  constant  fraction  of  |th  of  a 
square  foot  for  every  foot  in  depth  of  the  plane  to  the  quantity 
/S  previously  determined,  which  empirical  method  he  found  to 
agree  nearly  with  the  results  of  Beaufoy's  experiments. 

The  resistances  of  curved  surfaces,  such  as  the  bows  of  ships, 
were  adverted  to,  the  method  of  treating  them  being  to  divide 
the  depth  of  immersion  into  several  horizontal  layers,  and  then 
again  into  a  number  of  straight  portions,  and  to  deal  with  each 
portion  as  a  separate  detached  plane,  according  to  the  preceding 
rules. 

The  question  of  friction  was  then  considered.  The  experi- 
ments of  Beaufoy  were  referred  to,  giving  0*339  Ib.  per  square 
foot  as  the  co-efficient  of  friction  for  a  plained  and  painted  sur- 
face of  fir,  moved  through  the  water  at  10  feet  per  second,  the 
law  of  increase  being  nearly  as  the  squares  of  the  velocities,  viz., 
the  1-949  power.  Mr.  Phipps  was,  however,  of.  opinion,  that  a 
surer  practical  guide  for  determining  the  coefficient  of  friction 
would  be,  by  considering  all  the  data  and  circumstances  of  a 
steam-ship  of  modern  construction,  moving  through  the  water 
at  any  given  speed.  The  actual  indicated  horse-power  of  the 
engines  being  given,  the  slip  of  the  paddles  being  known,  and  the 
friction  and  other  losses  of  power  approximated  to,  it  was  clear 
that  the  portion  of  the  power  necessary  to  overcome  the  resistance 
of  the  vessel  might  be  easily  deduced.  By  determining  approxi- 
mately, by  the  preceding  rules,  the  amounts  of  the  Plus,  the 
Minus,  and  the  Additional  Head  resistances,  and  deducting  them 
from  the  total  resistance,  the  remainder  would  be  the  resistance 


438  STEAM    NAVIGATION. 

due  to  the  friction  of  the  surface.  By  this  process,  and  taking  as 
an  example,  the  iron  steam-ship  '  Leinster,'  when  perfectly  clean, 
and  going  on  her  trial  trip  30  feet  per  second  in  sea-water,  her  im- 
mersed surface  being  13,000  square  feet,  the  coefficient  of  fric- 
tion came  out  at  4*34  Ibs.  per  square  foot.  Beaufoy's  coefficient 
of  0*339  Ib.  per  square  foot  at  10  foot  per  second  would,  according 
to  the  square  of  the  velocities,  amount  to  3'051  Ibs.  at  30  feet  per 
second.  The  difference  between  this  amount  and  the  above  4*34 
Ibs.  might  be  accounted  for  by  a  difference  in  the  degree  of 
roughness  of  the  surfaces. 

Other  methods  for  the  determination  of  the  coefficient  of 
friction  were  then  discussed.  One,  derived  from  the  known 
friction  of  water  running  along  pipes,  or  water-courses,  was 
shown  to  be  considerably  in  excess  of  the  truth.  It  was  founded 
upon  the  observed  fact,  that  at  a  velocity  of  15  feet  per  second,  the 
friction  of  fresh  water  on  the  interior  of  a  pipe  was  25  oz.*  per 
square  foot.  Apply  ing  this  to  the  ship  '  Leinster,'  and  increasing 
the  friction  as  the  square  of  the  velocities  up  to  30  feet  per 
second,  the  above  friction  would  become  100  oz.,  or  6£  Ibs.,  per 
square  foot,  which,  acting  upon  13,000  square  feet  of  surface, 
would  absorb,  at  the  above  speed,  no  less  than  4,395  H.P.,  whilst 
the  total  available  power  of  the  engines  (after  deducting  from  the 
indicated  4,751  H.P.  ^th  for  friction,  working  air-pumps,  and 
other  losses,  and  |th  of  the  remainder  for  the  observed  slip),  was 
only  3,421  H.P. ;  thus  showing  an  excess  of  resistance  equal  to 
974  H.P.,  without  allowing  any  power  to  overcome  the  other  re- 
sistances. The  assumption  of  25  oz.  being  the  proper  measure 
of  the  friction  per  square  foot,  at  a  velocity  of  15  feet  per  second, 
upon  the  clean  surface  of  an  iron  ship,  seemed  to  have  arisen 
from  the  opinion,  very  generally  entertained,  that  there  was  no 
difference  in  the  amount  of  friction  in  pipes  and  water-courses, 
whether  internally  smooth  like  glass,  or  moderately  rough  like 
cast-iron,  and  that  the  surfaces  of  ships  were  subject  to  the  same 
action.  The  comparatively  recent  experiments,  in  France,  of  the 
late  M.  Henry  Darcy  were  in  opposition  to  the  above  view,  and 

*  For  sea  -water  this  quantity  must  be  increased  as  the  specific  gravity,  or  as 
62-5  to  M-2. 


EFFECT    OF    SMOOTHNESS    OF    SUKFACE.  439 

showed  that  the  condition  as  to  roughness  of  the  interior  of  a 
pipe  modified  the  friction  considerably.  Thus,  with  three  differ- 
ent conditions  of  surface,  the  coefficients  were : 

A.  Iron  plate  covered  with  bitumen  made  very  smooth,  Q'000432 

B.  New  cast-iron 0-000584 

C.  Oast-iron  covered  with  deposits       .        .        .        .0-001167 
The  friction  was,  therefore,  nearly  as  1,  1£,  and  3. 

As  there  appeared  no  reason  to  doubt  the  correctness  of  M. 
Darcy's  experiments,  even  in  pipes  the  notion  of  the  friction  being 
uninfluenced  by  the  state  of  roughness  of  the  interior  could  no 
longer  be  entertained.  The  25  oz.,  previously  mentioned  as  the 
measure  of  friction  per  square  foot  for  the  interior  of  pipes  and 
water-courses,  could  not,  therefore,  be  regarded  as  a  constant 
quantity,  applicable  to  all  kinds  of  surfaces;  but  from  Mr. 
Phipps'  calculations,  it  appeared  to  come  intermediately  between 
the  coefficients  of  the  surface  B  and  0,  given  in  the  above  scale ; 
as  at  15  per  second, 

A  would  give  13^  oz.  per  square  foot 
B  "         20  "  " 

and  0  "         40  "  " 

Besides,  there  was  another  cause  for  an  excess  of  friction  in 
pipes  and  water-courses,  over  that  upon  ships,  even  when  the 
surfaces  were  equally  smooth.  It  arose  from  the  circumstance, 
that  where  the  velocity  of  the  water  hi  a  pipe,  or  open  water- 
course, was  spoken  of,  the  meaning  was,  its  average  velocity; 
whilst  the  velocity  of  a  vessel  through  still  water  meant  what  the 
words  implied,  namely,  the  relation  of  the  vessel's  motion  to  the 
fluid  at  rest.  If  the  case  were  taken  of  a  water-course  of  such 
width,  that  the  friction  of  the. bottom  only  need  be  considered, 
with  an  average  velocity  of  flow  of  15  feet  per  second,  the  friction 
upon  the  bottom  would  be  equal  to  25  oz.  per  square  foot ;  but 
according  to  the  rules  generally  used,  an  average  velocity  of  15 
feet  per  second  corresponded  to  a  surface  velocity  of  16-66  feet 


440  STEAM   NAVIGATION. 

per  second,  which  was  the  velocity  with  which  a  vessel  should 
pass  through  still  water,  to  give  an  equal  friction  upon  its  sides. 
According  to  Beaufoy,  the  velocity  of  16-66  feet  per  second  would 
produce  a  friction  of  -932  Ibs.  or  14-91  oz.,  where  15  feet  would 
only  give  12'2  oz.  The  difference  between  14*91  oz.  and  25  oz. 
(equal  to  10*09  oz.)  must,  therefore,  Mr.  Phipps  thought,  be  set 
down  to  the  different  degree  of  roughness  of  the  surfaces  in  the 
water-course  and  the  vessel. 

Taking  then  4*34  Ibs.  as  the  friction  per  square  foot  of  a  new 
iron  ship,  moving  through  the  water  at  a  speed  of  30  feet  per 
second,  it  would  be  found,  Mr.  Phipps  considered,  that  this  was 
equal  to  the  o-g4-<iT  Par^  °f  ^e  P^us  resistance  of  a  plane  1  foot 
square,  moving  through  the  water  at  right  angles  to  itself  at  the 
above  velocity.  Also,  as  the  resistance  of  both  planes  increased 
according  to  the  same  law  of  the  square  of  the  velocities,  the 
ratio  of  1  to  207*06  would  subsist  at  all  velocities. 

64*2*»2  1 

The  ratio  was  -^-  to  4*34  Ibs.  =  ^^ 

Calling  the  ratio  r,  and  the  whole  frictional  surface  in  square  feet 
s,  and  ,#,  as  before,  the  area  of  a  square-acting  plane  of  equiv- 
alent resistance,  then 

S=s-r-r  =  s-s-  207-06. 

As  an  example  of  the  application  of  the  previous  deductions, 
the  performance  of  the  steam-ship  'Leinster,'  on  her  trial  trip, 
when  going  through  sea-water  at  a  speed  of  30  feet  per  second, 
was  referred  to. 

In  this  case — 

•TO,  the  area  of  the  immersed  midship  section  was  336  sq.  ft. 

d,  the  draught  of  water 13  ft. 

r,  the  reduced  ratio  of  the  slant  length  of  the  bow 

to  the  projection 10  to  1. 

r',  the  same  for  the  stern  ....  10  to  1. 
r",  the  ratio  of  1  square  foot  of  square-acting 

plane,  to  1  square  foot  of  frictional  surface        207'06  to  1. 

c,  the  velocity  in  feet  per  second          ...  30 


EXAMPLE    OF    STEAMER    '  LEINSTER.'  441 

w,  the  weight  of  a  cubic  foot  of  sea- water        .  64'21bs. 

f,  the  area  of  the  frictional  surface      .         .         .      13,000  sq.  ft- 

Calling  P,  the  Plus,  or  head  resistance ;  JLf,  the  Minus,  or  stern 
resistance ;  A,  the  Additional  Head  resistance ;  F,  the  Friction- 
al, or  surface  resistance ;  <Si  the  area  of  a  square-acting  plane 
having  an  equal  resistance  with  each  of  the  above ;  and  _Z?,  the 
total  resistance ; 

Then,  P-    ~        =S=     ~=     3-36  sq.ft. 

«     M=&       =S=^=     1-68       « 

r*  -  100 

1 3 
"     A=    £       =S=  1-86      " 

7 

"  '=&*-         «™   " 

u 

S=  69-68      " 

64-20* 

R  =  69-68  x =  69-68  x  900  =  62,712  Ibs 

80 

Iba. 
H  (Eealized  Power)  =  62,712  x  30  -*-  550  =  3420-66  H.P. 

H'  (Gross  Power)  including  the  slip  and  other  losses,  = 
8420-66  x  ~  =  4751  H.P. 

Thus,  by  ascertaining  the  value  of  S  for  any  vessel,  which 
was  entirely  independent  of  velocity,  it  would  be  easy  to  deter- 
mine the  power  necessary  to  propel  it  at  any  required  speed,  or 
the  speed  being  given,  to  find  the  corresponding  power. 

Generally  H=VS ^i|_Z!  ^.550  (!) 

Or,  because  for  sea  water  64-2  was  very  nearly  equal  to  2  gr, 


19* 


64-2 


442  STEAM   NAVIGATION. 

When  the  slip  and  other  losses  were  in  the  same  proportion  as 
in  the  '  Leinster ' : 

,  _      100 

When  the  gross  power  was  given,  and  the  velocity  was  required  ; 
/  72 

lwH'  x  55° 

t  F= 

\ 

Mr.  Phipps  then  proceeded  to  examine  the  question  of  the 
influence  of  form  in  reducing  the  resistance  of  vessels. 

It  was  argued  that,  in  vessels  of  similar  type  to  the  *  Lein- 
ster,' where  T\ths  of  the  whole  resistance  was  due  to  friction, 
and  only  ynth  to  considerations  involving  the  question  of  'form,' 
no  minor  modifications  of  the  latter  could  have  much  effect  in 
diminishing  the  total  resistance.  The  case  of  other  vessels  of 
different  type,  more  bluff  in  the  bows  and  not  so  fine  in  the  run, 
was  adverted  to,  and  a  particular  instance  was  discussed,  where 
the  inertial  resistance  was  supposed  to  be  equal  to  |th  of  the 
total  resistance,  and  the  slant  length  of  the  bows  to  the  base  to 
be  as  6  to  1.  If  such  a  vessel  were  altered,  so  as  to  make  the 
above  proportion  8J  to  1,  the  improvement  would  only  diminish 
the  total  resistance  by  TVth. 

The  conclusion  that  the  friction  of  ships  constitutes  the 
largest  part  of  their  resistances,  was  first  pressed  upon  me  in 
1854,  in  which  year  I  built  two  steamers  with  water-lines  formed 
on  the  principle  of  imparting  to  the  particles  of  water  the  mo- 
tion of  a  pendulum,  as  already  explained.  I  found,  as  I  expected, 
that  these  vessels  passed  through  the  water  with  great  smooth- 
ness, and  without  in  any  measure  raising  the  water  in  a  wave  at 
the  bow,  as  was  a  common  practice  in  the  older  class  of  steam- 
boats. Nevertheless  I  did  not  obtain  a  speed  much  superior  to 

*  If  for  fresh  water  H'  x  0-9T  =  Gross  H.P. 
t  If  for  fresh  water  V -t-  0-99  =  Velocity. 


MODES    OF   PREDICTING   VELOCITY.  443 

that  of  vessels  less  artistically  formed  ;  and  the  conclusion  be- 
came inevitable — seeing  that  all  other  known  causes  of  resist- 
ance had  been  reduced  to  a  minimum  without  material  benefit 
to  the  speed — that  the  friction,  which  alone  remained  unchanged, 
must  constitute  the  main  element  of  resistance;  and  other  things 
being  alike,  the  friction  of  a  vessel,  as  of  a  river,  would,  in  such 
case,  be  measurable  by  the  wetted  perimeter  of  the  cross-section. 
It  was  further  plain,  that  as  there  was  not  much  difference  be- 
tween the  resistance  of  a  vessel  formed  with  pendulum  or  wave 
curves,  and  that  of  well-formed  vessels  of  the  ordinary  configur- 
ation, any  mode  of  computing  the  resistance  applicable  in  the 
one  case  would  also  be  applicable  without  material  error  in  the 
other.  These  conclusions,  which  I  published  in  my  '  Catechism 
of  the  Steam-Engine,'  in  1856,  are  now  very  generally  accepted ; 
and  when,  in  1857,  Mr.  Rankine  had  to  compute  the  probable 
speed  of  an  intended  vessel,  he  proceeded  on  the  supposition 
that  the  resistance  was  due  almost  wholly  to  friction,  and  that 
the  friction  of  a  riband  of  the  form  of  a  trochoid  or  rolling 
wave,  of  the  length  of  the  ship  and  of  the  breadth  of  the  wet- 
ted perimeter,  would  be  an  accurate  measure  of  the  resistance, 
the  trochoid  being  the  same  order  of  curve  as  that  which  would 
be  described  by  a  pendulum.  Since,  however,  a  wave  moves  in 
different  parts  with  different  velocities,  Mr.  Rankine  concluded 
that  it  would  be  proper  to  take  this  circumstance  into  account, 
and  he  therefore,  instead  of  taking  the  actual  surface  of  the  ves- 
sel, took  a  surface  so  much  larger,  that  its  friction  would  pro- 
duce a  resistance  equivalent  to  the  increased  friction  caused  by 
the  varying  velocities  of  the  wave,  and  the  hydrostatic  pressure 
consequent  upon  the  difference  of  level  at  the  bow  and  stern, 
and  which  in  a  well-formed  vessel  is  very  small.  This  additional 
or  hypothetical  surface  Mr.  Rankine  terms  augmented  surface; 
and  by  using  this  theoretical  surface  in  his  computations  instead 
of  the  actual  wetted  surface  of  the  ship,  he  deduces  results  sin- 
gularly conformable  to  those  obtained  by  actual  experiment. 
The  amount  of  the  augmented  surface  will  vary  with  the  sharp- 
ness of  the  vessel — sharp  vessels  having  the  least  augmentation ; 


444 


STEAM   NAVIGATION. 


and  the  sharpness  is  measured  by  the  sines*  of  the  angles  of  the 
water-lines  at  the  bow  and  stern.  I  shall  here  introduce  Mr. 
Bankine's  able  investigation,  to  which  the  only  exception  that 
can  be  taken,  so  far  as  I  see,  is  that  the  resistance  per  square 
foot  produced  by  friction  in  every  part  of  the  length  of  the  ves- 
sel is  not  the  same,  but  is  more  at  the  fore  part,  in  consequence 
of  the  necessity  of  putting  the  water  into  motion ;  but  after  this 
has  been  done,  the  friction  per  square  foot  of  the  further  length 
of  the  vessel  will  be  uniform. 

The  Resistance  due  to  Frictional- Eddies  remains  alone  to  be  con- 
sidered. That  resistance  is  a  combination  of  the  direct  and  indirect 
effects  of  the  adhesion  between  the  skin  of  the  ship  and  the  particles  of 
water  which  glide  over  it ;  which  adhesion,  together  with  the  stiffness 
of  the  water,  occasions  the  production  of  a  vast  number  of  small  whirls, 
or  eddies,  in  the  layer  of  water  immediately  adjoining  the  ship's  sur- 


*  A  sine  is  one  of  the  measures  of  an  angle.  Thus  In  the  cir- 
cle A  D  c  E  (fig.  58)  the  lines  A  B  and  A  E  are  radii  of  the  circle  at 
right  angles  with  one  another,  and  o  a  is  the  sine  of  the  angle 
c  B  A,  and  D  a  Is  the  sine  of  the  angle  DBA.  The  circle  is  sup- 
posed to  be  divided  into  360  degrees,  so  that  a  quadrant,  or  one- 
fourth  of  a  circle,  is  90  degrees. 

In  fig.  59  the  various  trigonometrical  quantities  relating  to 
the  angle  A  are  graphically  represented.  The  angle  A  is  half  a 
right  angle,  or  45  degrees,  which  is  the  eighth  part  of  the  whole 
circle  of  860  degreea 

Fig.  59. 


;. COSfNE- -3 


<"- "*— — RA  D I U  S  ;-• 


RESISTANCE    FROM   FRICTIONAL    EDDIES.  445 

face.  The  velocity  with  which  the  particles  of  water  whirl  in  those  ed- 
dies, bears  some  fixed  proportion  to  that  with  which  those  particles 
glide  over  the  ship's  surface;  hence  the  actual  energy  of  the  whirling 
motion  impressed  on  a  given  mass  of  water  at  the  expense  of  the  pro- 
pelling power  of  the  ship,  being  proportional  to  the  square  of  the  velocity 
of  the  whirling  motion,  is  proportional  to  the  square  of  the  velocity  of 
gliding ;  in  other  words,  it  is  proportional  to  the  height  due  to  the  ve- 
locity of  gliding.  The  velocity  of  gliding  of  the  particles  of  water  over 
a  given  portion  of  the  ship's  skin,  bears  a  ratio  to  the  speed  of  the  ship 
depending  on  her  figure,  and  on  the  position  of  the  part  of  her  skin  in 
question  ;  and  the  height  due  to  the  velocity  of  gliding  is  equal  to  the 
height  due  to  the  speed  of  the  ship,  multiplied  by  the  square  of  the  same 
ratio.  Further,  the  mass  of  water  upon  which  whirling  motion  is  im- 
pressed by  a  given  part  of  the  ship's  skin  while  she  advances  through  a 
unit  of  distance,  is  proportional  to  the  area  of  that  part  of  the  skin,  mul- 
tiplied by  the  before-mentioned  ratio  which  the  velocity  of  gliding  of  the 
water  past  that  part  of  the  skin  bears  to  the  velocity  of  the  ship. 

Hence  the  resistance  to  the  motion  of  the  ship,  due  to  the  production  of 
f notional  eddies  ~by  a  given  portion  of  Tier  skin,  is  the  product  of  the  fol- 
lowing factors : — 

I.  The  area  of  the  portion  of  the  ship's  skin  in  question. 

II.  The  cube  of  the  ratio  which  the  velocity  of  gliding  of  the  particles 
of  water  over  that  area  bears  to  the  speed  of  the  ship  ;  being  a  quantity 
depending  on  the  figure  of  the  ship  and  the  position  of  the  part  of  her 
skin  under  consideration. 

III.  The  height  due  to  the  ship's  speed ;  that  is, 

(speed  in  feet  per  second)2 

6?i 

(speed  in  knots)8 
or, — 

22-6 

IV.  The  heaviness  (or  weight  of  a  unit  of  volume)  of  the  water 
(64  Ibs.  per  cubic  foot  for  sea-water). 

V.  A  factor  called  the  coefficient  of  friction,  depending  on  the  mate- 
rial with  which  the  ship's  skin  is  coated,  and  its  condition  as  to  rough- 
ness or  smoothness. 

The  sum  of  the  products  of  the  Factors  I.  and  II.  for  the  whole  skin 
of  the  ship  has  of  late  been  called  her  AUGMENTED  SURFACE  ;  and  the 
Eddy-resistance  of  the  whole  ship  may  therefore  be  expressed  as  the 
product  of  her  Augmented  Surface  by  the  Factors  III.  IV.  and  V.  above 
mentioned.* 


*  In  algebraical  symbols,  let  d  <  denote  the  area  of  a  small  portion  of  the  ship's 


446  STEAM   NAVIGATION. 

The  resistance  thus  determined,  being  deduced  from  the  work  per- 
formed in  producing  eddies,  includes  in  one  quantity  both  the  direct  ad- 
hesive action  of  the  water  on  the  ship's  skin,  and  the  indirect  action, 
through  increase  of  pressure  at  the  bow  and  diminution  of  the  pressure 
at  the  stern. 

The  existence  of  this  kind  of  resistance  has  been  recognised  from  an 
early  period.  Beaufoy  made  experiments  on  models  to  determine  its 
amount ;  Mr.  Hawksley  and  Mr.  Phipps  have  included  it  in  a  formula  for 
the  resistance  of  ships ;  and  Mr.  Bourne  pointed  out  that  it  must  depend 
mainly  on  the  ship's  immersed  girth.  But  the  earlier  researches,  both 
experimental  and  theoretical,  throw  little  light  on  the  subject,  and  fail 
to  give  a  trustworthy  value  of  the  coefficient  of  friction ;  because  in 
them  it  was  assumed  that  the  frictional  resistance  was  proportional  to 
the  actual  immersed  surface  of  the  vessel,  and  the  variations  of  the 
speed  of  the  gliding  of  the  water  over  different  parts  of  that  surface 
were  neglected. 

When  the  Editor  of  this  treatise  t  (having  occasion  to  compute,  in 
1857,  the  probable  resistance  at  a  given  speed  of  a  steam -vessel  built  by 
Mr.  J.  E.  NapierJ,  introduced  for  the  first  time  the  consideration  of  the 
augmented  surface,  he  adopted,  for  the  coefficient  of  friction,  the  con- 
stant part  of  the  expression  deduced  by  Professor  Weisbach  from  exper- 
iments on  the  flow  of  water  in  iron  pipes,  viz. : 


and  that  value  has  given  results  corroborated  by  practice,  for  surfaces 
of  clean  painted  iron.  For  clean  copper  sheathing,  and  for  very  smooth 
pitch,  it  appears  probable  that  the  coefficient  of  friction  is  somewhat 
smaller;  but  there  are  not  sufficient  experimental  data  to  decide  that 
question  exactly.  Experimental  data  are  also  wanting  to  determine 
the  precise  increase  of  the  coefficient  of  friction  produced  by  various 
kinds  and  degrees  of  roughness  and  foulness  of  the  ship's  bottom ;  but 
it  is  certain  that  that  increase  is  sometimes  very  great. 

The  preceding  value  of  the  coefficient  of  friction  leads  to  the  follow- 
ing very  simple  rule  for  clean  painted  iron  ships: — At  ten  knots,  theeddy- 


skin ;  ?,  the  ratio  •which  the  velocity  of  gliding  of  the  water  over  that  portion 
bears  to  the  speed  of  the  ship ;  c,  the  speed  of  the  ship ;  y,  gravity ;  w,  the  heavi- 
ness of  the  water ;  /,  the  coefficient  of  friction ;  then 

Eddy-resistance  ^fw—fq*  ds; 

%t 
/q3  d  s  being  the  Augmented,  Surface. 

t  The  treatise  referred  to  is  a  '  Treatise  on  Shipbuilding,'  by  Mr.  Kankino  and 
other  eminent  authorities,  in  course  of  publication  in  1865. 


COMPUTATION    OP    POWER   AND    SPEED.  447 

resistance  is  one  pound  avoirdupois  per  square  foot  of  augmented  surface  ; 
and  varies,  for  otlwr  speeds,  as  the  square  of  the  speed, 

COMPUTATION    OP    PROPELLING    POWER    AND    SPEED. 

General  Explanations. — The  method  of  calculation  now  to  be  ex- 
plained and  illustrated  was  first  practically  used  in  1857,  under  the  cir- 
cumstances stated.  A  very  condensed  account  of  it,  illustrated  by  a 
table  of  examples,  was  read  to  the  British  Association  in  September, 
1861,  and  printed  in  various  mechanical  journals  for  October  of  that 
year;  and  some  further  explanations  appeared  in  a  paper  on  Waves  in 
the  'Philosophical  Transactions  for  1862.'* 

The  method  proceeds  by  deducing  the  eddy -resistance  from  an  ap- 
proximate value  of  the  augmented  surface.  It  is  therefore  applicable  to 
those  vessels  only  in  which  eddy-resistance  forms  the  whole  of  the  ap- 
preciable resistance ;  but  such  is  the  case  with  all  vessels  of  proportions 
and  figures  well  adapted  to  their  speed,  as  has  been  explained  in  the 
preceding  sections  ;  and  as  for  misshapen  and  ill-proportioned  vessels, 
there  does  not  exist  any  theory  capable  of  giving  their  resistance  by  pre- 
vious computation. 

Computation  of  Augmented  Surface. — To  compute  the  exact  aug- 
mented surface  of  a  vessel  of  any  ordinary  shape  would  be  a  problem 
of  impracticable  labour  and  complexity.  The  method  employed,  there- 
fore, as  an  approximation  for  practical  purposes,  is  to  choose  in  the 
first  instance  a  figure  approximating  to  the  actual  figure,  but  of  such  a 
kind  that  its  augmented  surface  can  be  calculated  by  a  simple  and  easy 
process,  and  to  nse  that  augmented  surface  instead  of  the  exact  aug- 
mented surface  of  the  ship ;  care  being  taken  to  ascertain  by  comparison 
with  experiments  on  ships  of  various  sizes  and  forms  whether  the  ap- 
proximation so  obtained  is  sufficiently  accurate. 

The  figure  chosen  for  that  purpose  is  the  trochoid,  or  rolling-wave- 
curve,  extending  between  a  pair  of  crests,  such  as  A  and  B  in  fig.  60 ; 

Fig.  60. 


for  by  an  easy  integration,  published  in  the  'Philosophical  Transactions 
for  1862,'  it  is  found  that  the  augmented  surface  of  a  trochoidal  riband  t 

*  A  prediction  of  the  speed  of  the  '  Great  Eastern,'  with  different  amounts  of 
engine-power,  obtained  by  this  method  of  calculation,  was  published  in  the 
•  Philosophical  Magazine '  for  April,  1859. 

t  This  is  the  species  of  curve  that  will  be  described  by  a  pendulum,  the  surfeoe 


448  STEAM   NAVIGATION. 

of  a  given  length  in  a  straight  line,  and  of  a  given  breadth,  is  equal  to 
the  product  of  that  length  and  breadth,  multiplied  by  the  following  co- 
efficient of  augmentation ; — 

1  +  4  (sine  of  greatest  obliquity)^  +  (sine  of  greatest  obliquity)4  ;  the 
greatest  obliquity  meaning  the  greatest  angle,  BED,  made  by  a  tangent, 
D  E,  to  the  riband  at  its  point  of  contrary  flexure,  D,  with  its  straight 
chord,  A  B. 

In  approximating  to  the  augmented  surface  of  a  given  ship  by  the 
aid  of  that  of  a  trochoidal  riband,  the  following  values  are  employed  : 

I.  For  the  length,  AB,  of  the  riband,  the  length  of  the  ship  on  the 
plane  of  flotation. 

II.  For  the  total  breadth  of  the  riband,  the  mean  immersed,  girth  ; 
found  by  measuring,  on  the  body-plan,  the  immersed  girths  of  a  series 
of  cross-sections,  and  taking  their  mean   by  Simpson's  Eule,  or  by 
measuring  mechanically  with  an  instrument  the  sum  of  a  number  of 
girths,  and  dividing  by  their  number. 

III.  For  the  coefficient  of  augmentation,  the  mean  of  the  values  of  that 
coefficient  as  deduced  from  the  greatest  angles  of  obliquity  of  the  series 
of  water-lines  of  the  fore-body,  shown  on  the  half-breadth  plan.     It  is 
not  necessary  to  measure  the  angles  themselves,  but  only  their  sines. 

The  augmented  surface  is  then  computed  by  multiplying  together 
those  three  factors. 

The  Computation  of  the  Probable  Resistance  (in  Ibs.)  at  a  given  speed 
is  performed  according  to  the  rule  already  stated,  by  multiplying  the 
augmented  surface  by  the  square  of  the  speed  in  knots,  and  dividing  by  100 
(for  clean  painted  iron  ships).  . 

The  process  just  described  is  virtually  equivalent  to  the  following: — 
An  ocean  wave  is  conceived  (A  c  B  in  fig.  60),  of  a  length,  A  B,  equal  to 
that  of  the  ship  on  her  water-line ;  and  having  its  steepest  angle  of 
slope,  BED,  such  that  the  function  of  that  slope,  given  in  Article  162  as 
the  coefficient  of  augmentation,  shall  be  equal  to  the  mean  value  of  the 
same  function  for  all  the  water-lines  of  the  ship's  bow.  A  solid  of  a 
breadth  equal  to  the  ship's  mean  immersed  girth  is  then  conceived  to  be 
fitted  into  the  hollow,  A  c  B,  and  to  be  moved  along  with  the  advance  of 
the  wave ;  and  the  resistance  due  to  frictional  action  between  that  solid 
and  the  particles  of  water  is  taken  as  the  approximate  value  of  the  re- 
sistance of  the  vessel. 

In  Computing  the  Probable  Engine  Power  required  at  a  given  Speed, 
allowance  must  be  made  for  the  power  wasted  through  slip,  through 
wasteful  resistance  of  the  propeller,  and  through  the  friction  of  the  en- 

of  which  was  shown  by  me  in  my  '  Catechism  of  the  Steam-Engine,'  published  in 
1856,  to  be  the  measure  of  the  resistance — a  conclusion  deduced  by  me  from  exper- 
iment several  years  before. 


RULE   FOR  COMPUTING   SPEED.  449 

gine.  The  proportion  borne  by  that  wasted  power  to  the  effective  or 
net  power  employed  in  driving  the  vessel,  of  course  varies  considerably 
in  different  ships,  propellers,  and  engines ;  but  in  several  good  examples 
it  has  been  found  to  differ  little  from  0'63 ;  so  that,  as  a  probable  value 
of  the  indicated  power  required  in  a  well-designed  vessel,  we  may  take — 

net  power  x  1-63. 

Now  an  indicated  horse-power  is  550  foot-pounds  per  second ;  and  a 
knot  is  1'688  foot  per  second;  therefore  an  indicated  horse-power  is 

550 

—  =  326  knot-pounds,  nearly ; 
1*688 

or  326  Ibs.  of  gross  resistance  overcome  through  one  nautical  mile  in  an 
hour.  If  we  estimate,  then,  the  net  or  useful  work  done  in  propelling 
the  vessel  as  equal  to  the  total  work  of  the  steam  divided  by  1'63,  we 
shall  have 

326 

— — -  =  200  knot-pounds 

1'63 

of  net  work  done  in  propulsion  for  each  indicated  horse-power.  Hence 
the  following 

RULE. — Multiply  the  Augmented,  Surface  in  square  feet  by  the  cube  of 
the  speed  in,  knots  and  divide  by  20000 ;  the  quotient  will  be  the  probable 
indicated  horse-power. 

The  divisor  in  this  rule,  20000,  expresses  the  number  of  square  feet  of 
augmented  surface  which  can  be  driven  at  one  knot  by  one  indicated 
horse-power  :  it  may  be  called  the  COEFFICIENT  OF  PROPULSION. 

It  is,  of  course,  to  be  understood  that  the  exact  coefficient  of  propul- 
sion differs  in  different  vessels,  according  to  the  smoothness  of  the  skin, 
the  nature  of  its  material,  and  the  efficiency  of  the  engines  and  propel- 
lers ;  being  greatest  in  the  most  favourable  examples. 

In  clean  iron  ships,  with  no  evident  fault  in  shape  or  dimensions,  or 
in  the  propeller  and  engine,  it  has  been  found  on  an  average  to  be  some- 
what above  20000 ;  and  the  value  20000  may  be  taken  as  a  probable  and 
safe  estimate  of  the  coefficient  of  propulsion  in  any  proposed  vessel  de- 
signed on  good  principles.  In  every  instance  in  which  that  coefficient 
is  materially  less  than  20000,  the  shortcoming  can  be  accounted  for  by 
some  fault,  such  as  undue  bluntness  of  the  bow  or  stern. 

In  vessels  sheathed  with  copper  or  coated  with  smooth  pitch,  the  co- 
efficient of  propulsion  is  unquestionably  greater ;  but  in  what  precise 
proportion  it  is  at  present  difficult  to  say,  owing  to  the  scarcity  of  ex- 
perimental data. 

Computation  of  Probable  Speed. — When  the  augmented  surface  of  a 
ship  has  been  determined,  her  probable  speed  with  a  given  power  is 
computed  as  follows : — 


450  STEAM   NAVIGATION. 

Multiply  tTie  indicated  Horse-power  by  tJte  Coefficient  of  Propulsion  (say 
for  clean  iron  ships,  20000) :  divide  by  the  Augmented  Surface,  and  extract 
the  cube  root  of  the  quotient  for  the  probable  speed  in  knots. 

EXAMPLE  I. — Calculation  of  Probable  Speed  of  H.  M.  S.  '"Warrior.' 

Displacement  on  Trial 8997  tons 

Draught  ofWater ]  *"*...  |;8|  feet 

Water-lines.                       Sine  of  Obliquity.              Square  of  Sine.  4th  power  of  Sine. 

L.W.L  -870  -1369     -01874 

2  W.L  -315  '0992     -00984 

3  W.L  -290  -0841     -00707 

4  W.L  -265  -0702     -00492 

5  W.L  -235  -0552     -00304 

6  W.L  -165  '0272     -00074 

Keel  -000  -0000     00000 

Means -0674     -00583 

1  +  (4  x  -0674)  +  -00583  -  1-275,  Coefficient  of  Augmentation. 

Half-girths  from  Body-plan                      Simpson's  Products 

Foot.                                    Multipliers.  iTOttncW. 

21-0        1        21-0 

27-2 4        108-8 

80-8        2        61-6 

34-0        4        138-4 

88-8        2        77.6 

41-5        4        160-0 

42-6        2          85-2 

44-0        4        176.0 

44-0        2        88-0 

44-0        4        176.0 

43-3        2        86-6 

42'1        4        168-4 

40-3        2        80-6 

381        4        152-4 

86-0        2        72-0 

85-0        4        140-0 

82-0        1        32-0 


Divideby 8)1830-6  Sum. 

Divide  by  i  number  of  Intervals. S)  6-012 

Mean  Immersed  Girth 76-8 

x  Length 380 

Product 28994 

x  Coefficient  of  Augmentation 1-275 

Augmented  Surface 86979  Square  Feet. 

Indicated  Horse-power  on  Trial 5471 

x  Coefficient  of  Propulsion 20000 

Divide  by  Aug.  Surface 36979)109,420,000  Product. 

Cube  of  Probable  Speed 2959 

Probable  Speed,  computed. 14-856  Knots. 

Actual  Speed,  on  Trial 14-354 

Difference ...  .    O02 


EXAMPLE    OF    COMPUTATION    OF    SPEED.  451 

EXAMPLE  II. — H.  M.  S.  'Fairy'  will  next  be  taken  as  an  example,  on 
account  of  the  great  contrast  in  size  between  her  and  the  '  Warrior.' 

Displacement 165  tons. 

Draught  of  Water 4 -S3  leet. 

Water-lines.  Sine  of  Obliquity.  Square  of  Sine.        Fourth  power  of  Sine. 

L.W.L     "23     -0529     -0028 

2  W.L      -22     -0484     -0023 

8  W.L      "21     -0441     -0019 

4  W.L      -17     -0289     -0008 

Keel         0     0     0 


Means -0304  '0015 

1  +  (4  x  -0304)  +  -0015  -  1-123,  Coefficient  of  Augmentation. 

Length  on  Water-line 144  Feet. 

x  Mean  Immersed  Girth  (measured  mechanically  with 

an  Instrument 19 

x  Coefficient  Augmentation 1-123 

Augmented  Surface 8072  Square  Feet 

Indicated  Horse-power,  on  Trial 364 

x  Coefficient  of  Propulsion 20000 


•+•  Augmented  Surface 3072)7,280,000  Product. 

Cube  of  Probable  Speed 2370 

Probable  Speed,  computed 13-383  Knots. 

Actual  Speed,  on  Trial 13-324 

Difference -009 

The  '  Fairy '  occurs  in  the  table  of  examples  given  in  the  paper  of 
1861,  already  referred  to  :  in  the  present  paper  the  measurements  have 
been  revised  and  improved  in  precision,  especially  as  regards  the  co- 
efficient of  augmentation.  The  difference  in  the  result  is  but  small. 

EXAMPLE  III. — H.  M.  S.  'Victoria  and  Albert' — a  wooden  vessel, 
sheathed  with  copper,  will  now  be  employed,  not  to  illustrate  the  com- 
putation of  probable  power  at  a  giwn  speed,  or  of  probable  speed  at  a 
given  power ;  but  to  compute  a  value  of  the  coefficient  of  propulsion 
for  a  copper-sheathed  vessel. 

Displacement  on  Trial  Trip 1980  tons. 

Draught  of  w»ter j  i^d;  ;;;}{$  fe^ 

Water-line*.  Sine  of  Obliquity.  Square  of  Sine.  4th  power  of  Sine. 

L.W.L      -19     -0861        -0018 

2  W.L     -185  -0842     -0012 

8  W.L      -17     -0289     -0008 

4  W.L     -14     -0196     -0004 

Keel         0    0    0 


Means -0252    -0008 

1  +  (4  x  -0252)  +  -0008  =  1-102,  Coefficient  of  Augmentation. 


452  STEAM   NAVIGATION. 


Length  on  Water-lino 800  Feet. 

xMean  Immersed  Girth  (measured  mechanically  with 

an  Instrument) 40 

x  Coefficient  of  Augmentation...  ..  1-102 


Augmented  Surface 13224  Square  Feet. 

x  L-ube  of  Speed  in  Knots IT3  =  4913 


*  Indicated  Horse-power  on  Trial 2980)64,969,512  Product. 


Coefficient  of  Propulsion 21,802 

Had  the  probable  speed  been  computed  with  the  coefficient  of  propulsion 
20000,  the  result  would  have  been  16-53  knots,  instead  of  17. 

Proportions  of  Length  to  Breadth. — Principles  which  have  been  al- 
ready explained  fix  the  least  absolute  length  suitable  for  a  vessel  which  is 
to  be  driven  at  a  given  speed.  But  after  that  least  length  has  been 
fixed,  a  question  may  arise  as  to  whether  that  least  length,  or  a  greater 
length,  is  the  most  economical  of  power.  That  question  is  answered  by 
finding  the  proportion  of  length  to  breadth,  which  gives  the  least  aug- 
mented surface  with  the  required  displacement. 

That  proportion  can  be  found  in  an  approximate  way  only ;  because 
of  the  approximate  nature  of  the  process  by  which  the  augmented  sur- 
face itself  is  found.  The  following  are  some  of  the  results  obtained  in 
certain  cases : — 

I.  When  the  proportion  of  breadth  to  draught  of  water,  and  the 
figure  of  cross-section,  are  fixed,  so  that  the  mean  girth  bears  a  fixed 
proportion  to  the  breadth,  it  appears  that  the  proportion  of  length  to 
breadth  which  gives  the  least  augmented  surface  for  a  given  displace- 
ment, is  about  7  to  1. 

II.  When  the  absolute  draught  of  water  is  fixed,  the  proportion  of 
length  to  breadth  which  gives  the  least  augmented  surface  for  a  given 
displacement  depends  on  the  proportion  borne  by  the  draught  of  water 
to  a  mean  proportional  between  the  length  and  breadth,  and  on  the 
figures  of  the  cross-sections.    The  following  are  some  examples  for  flat- 
bottomed  vessels : — 

( -I/Length  x  Breadth)  , 

--  from  4  to  5;  7  to  10;  12to  16;  17  to  23; 


Draught 
Length 


10. 


Breadth 

HI.  By  cutting  a  vessel  in  two  amidships,  and  inserting  a  straight 
middle  body,  the  proportion  borne  by  her  resistance  to  her  displace- 
ment is  always  diminished ;  because  the  midship  section  has  a  less 
mean  girth  in  proportion  to  its  area  than  any  other  cross-section  of  the 
ship ;  and  therefore  the  new  middle  body  adds  proportionally  less  to 
the  augmented  surface  than  it  does  to  the  displacement. 

IV.  It  does  not  follow,  however,  that  a  straight  middle  between 
tapering  ends  is  the  most  economical  form ;  for  by  adopting  continuous 


SUMMARY    OF    MAIN    DOCTRINES.  453 

curves  from  bow  to  stern  for  the  water-lines,  instead  of  the  lines  com- 
pounded of  curved  ends  and  a  straight  middle,  the  same  length,  the 
same  displacement,  and  almost  exactly  the  same  mean  girth  may  be 
preserved,  and  the  obliquity  of  the  water-lines  at  the  entrance  dimin- 
ished. 

GENERAL  CONCLUSIONS. 

The  principal  conclusions  to  be  drawn  from  the  foregoing  ex- 
position are  the  following : — 

1st.  That  the  bulk  of  a  ship  should  be  equal  to  half  the  bulk  of 
the  circumscribing  parallelepiped,  supposing  the  areas  of  all  the 
cross-sections  have  been  translated  into  the  form  of  a  rectangle. 

2d.  That  the  sectional  area  of  each  successive  frame  should 
vary  as  the  square  of  the  distance  from  the  stem  or  stern,  until 
the  points  midway  between  the  midship  frame  and  the  stem  or 
stern  have  been  reached,  and  that  the  areas  at  all  the  frames 
should  vary  in  the  manner  already  pointed  out. 

3d.  That  it  is  better  to  place  the  midship  frame  before  the 
centre  of  the  ship,  in  order  that  any  wave  raised  at  the  stern 
may  be  sufficiently  far  forward  to  assist  the  propulsion. 

4th.  That  the  horizontal  water-lines  should  be  pendulum  or 
trochoidal  curves,  or  such  equivalent  curve  as  will  enable  the 
progressive  displacement  to  follow  the  prescribed  law,  and -that 
the  transverse  section  should  be  formed  with  similar  curves  made 
as  nearly  as  possible  coincident  with  a  semicircle. 

5th.  That  nearly  the  whole  of  the  resistance  in  a  well-formed 
vessel  is  made  up  of  friction,  and  that  the  friction  per  square 
foot  of  surface  is  less  at  the  stern  than  at  the  bow,  but  that  the 
law  of  variation  is  not  known.  Also,  that  at  a  certain  point  of 
the  length  the  water  adhering  to  the  ship  will  attain  its  maxi- 
mum velocity,  and  thereafter  every  foot  of  the  length  will  have 
the  same  resistance. 

6th.  That  the  friction  is  diminished  by  making  the  bottom 
fair  and  smooth,  and  by  coating  it  with  a  suitable  lubricant,  and 
that  a  portion  of  the  power  expended  in  friction  may  be  recov- 
ered by  making  the  stern  part  of  the  vessel  to  overhang  near  the 
water-line,  so  as  to  be  propelled  by  the  upward  motion  of  the 
current  which  the  friction  generates,  and  also  by  placing  the 
propeller  in  the  stern  or  quarters  instead  of  at  the  sides 


454 


STEAM   NAVIGATION. 


TtL.  That  both  by  Boulton  and  Watt's  method  and  by  Mr. 
Eankine's  method  the  speed  of  a  steamer  may  be  accurately  pre- 
dicted. Boulton  and  Watt,  by  whom  the  screw  engines  of  the 
'  Great  Eastern '  were  made,  predicted  that  the  speed  of  the  ves- 
sel would  be  16'57  statute  miles  with  10,000  actual  horse-power. 
"Not  more  than  8,000  horse-power  were  actually  generated,  in 
consequence  of  a  deficiency  of  steam.  But  on  trial  the  speed 
was  found  to  be  as  nearly  as  possible  what  it  ought  to  be  accord- 
ing to  their  rule  with  this  proportion  of  power.  The  coefficient 
they  employed  for  statute  miles  in  making  this  computation  was 
900,  which  is  also  the  coefficient  they  habitually  use  in  the  case 
of  fast  river  boats  of  considerable  size  and  good  form. 

8th.  That  any  expedient  for  diminishing  the  resistance  of 
well-formed  vessels  to  be  of  material  efficacy  must  have  for  its 
object  the  diminution  of  the  friction  of  the  bottom,  either  by  re- 
ducing the  adhesion  of  the  particles  of  water  to  the  ship,  or  to 
one  another,  or  both  ;  and  also  by  making  the  adhering  surface 
as  small  as  possible. 

EXAMPLES  OF  LINES  OF  APPROVED  STEAMERS. 

In  fig.  61  we  have  the  body-plan  of  H.  M.  screw  yacht  'Fai- 
ry,' 144  feet  8  inches  long  between  perpendiculars ;  and  the  hor- 

Fig.  61. 


BODY  PLAN  OP  H.M.S.  '  FAIRY.' 

izontal  water-lines  can  easily  be  constructed  from  the  body-plan, 
by  dividing  the  length  by  the  number  of  vertical  lines  or  frames, 
and  by  setting  off  at  each  division  the  given  breadth  of  each  wa- 


STEAMERS    '  RATTLER '    AND    '  BREMEN.' 


455 


ter  line  at  that  part.  The  '  Fairy  '  is  312  tons,  21  feet  1|  inch 
extreme  breadth,  and  has  74*4  square  feet  of  immersed  section  at 
5  feet  draught.  She  is  propelled  by  two  oscillating  geared  en- 
gines of  42  inches  diameter  of  cylinder  and  3  feet  stroke,  and  has 
attained  a  speed  of  13'3  knots  per  hour,  exerting  363'8  indicated 
horse-power. 

The  'Rattler'  is  176  feet  6  inches  long  between  perpendicu- 
lars, 32  feet  8|  inches  extreme  breadth,  888  tons  burden,  894 
tons  displacement  at  11  feet  Scinches  mean  draught,  281-8  square 

Fig.  62. 


BODT  FLAX  OF  H.  II.  S.  '  RATTLER.' 

feet  of  immersed  section,  and  is  propelled  by  geared  engines  of 
200  nominal  horse-power.  With  428  indicated  horse-power  she 
attained  a  speed  of  10  knots — a  high  result,  imputable  partly  to 
her  good  form  for  such  speed,  and  partly  to  the  smoothness  of 
the  copper  sheathing — the  'Rattler'  being  a  wooden  vessel. 

In  the  steamer  '  Bremen,'  which  has  given  a  very  favourable 
result  in  working,  the  length  of  keel  and  fore-rake  is  318  feet; 
the  breadth  of  beam  40  feet;  depth  of  hold  26  feet;  tonnage, 
builder's  measurement,  2,500  tons ;  power,  two  direct-acting  in- 
verted cylinder  engines,  with  cylinders  of  90  inches  diameter 
and  &J  feet  stroke.  With  a  draught  of  18£  feet  the  displace- 


456 


STEAM    NAVIGATION. 


ment  was  3,440  tons,  the  area  of  immersed  section  606  square 
feet,  and  with  the  engines  working  to  1,624  horse-power  the 
speed  attained  was  13-15  knots. 

Fig.  64  is  the  body  plan  of  the  Cunard  steamer  'Persia.' 
The  vertical  sections  are  17-J-  feet  from  one  another,  and  the 
breadth  of  the  vessel  is  45  feet.  The  engines  are  side  lever ; 
cylinders  100  inches  diameter  and  10  feet  stroke,  making  18 
strokes  per  minute.  The  daily  consumption  of  coals  in  eight 
boilers  containing  40  furnaces  is  130  tons,  and  the  pressure  of 

Fig.  63. 


BODY  PLAN  OF  SCEKW  STEAMER  '  BREMEN.' 

the  steam  is  25  Ibs.  per  square  inch.  The  performance  of  the 
'  Persia '  has  been  very  satisfactory,  except  that  she  was  not 
strong  enough  in  the  deck  and  had  to  be  strengthened  there,  and 
she  has  a  great  deal  too  much  iron  in  the  shape  of  frames,  which 
conduce  to  weakness  rather  than  to  strength.  The  paddle- 
wheels  are  40  feet  in  diameter,  and  the  floats  are  10  feet  long 
and  3  feet  wide. 

In  fig.  65  the  body  plan  of  the  iron-plated  steamer  '  Warrior ' 
is  given,  and  66  is  a  transverse  section  of  the  same  vessel,  show- 
ing the  guns.  The  'Warrior'  is  an  iron-clad  steamer  of  6,039 
tons,  380  feet  long,  58  feet  broad,  and  1,250  horse-power ;  and 
with  the  exertion  of  5,460  actual  horse-power,  and  at  26  feet 
draught,  and  with  an  area  of  immersed  section  1,219  square  feet, 


STEAMERS    '  PERSIA  '    AND    '  WARRIOR.'  457 

she  realized  a  speed  of  14'3  knots.  The  utility  of  such  vessels  as 
the  '  "Warrior '  does  not  promise  to  be  considerable,  and  in  fact 
the  whole  idea  of  constructing  ships  that  would  be  impenetrable 
to  shot  turns  out  to  be  a  complete  delusion,  as  was  plainly  per- 
ceived by  a  number  of  competent  observers  would  necessarily  be 

Fig.  64. 


BOOT  PLAN  OF  PADDLE  STEAMER  '  PERSIA/ 

the  case  before  the  expensive  demonstrations  were  resorted  to 
which  the  Admiralty  has  thought  fit  to  institute.  If  there  had 
been  any  natural  law  which  restricted  the  penetrating  power  of 
ordnance  to  the  narrow  limits  hitherto  existing,  there  would 
have  been  some  reason  iu  the  conclusion  that  by  making  the  iron 
sides  of  ships  very  thick  the  shot  would  have  been  prevented 
20 


458 


STEAM   NAVIGATION. 


from  penetrating  them.  But  two  facts  were  quite  well  known  : 
first,  that  a  steel  punch  may  be  made  to  pierce  an  iron  plate  how- 
ever thick,  if  the  diameter  of  the  punch  he  equal  to  the  thick- 
ness of  the  iron ;  and,  secondly,  that  by  increasing  the  dimen- 
sions of  the  gun  an  amount  of  projectile  force  could  be  obtained 
that  would  suffice  for  the  punching  through  of  any  thickness 
whatever.  The  amount  of  this  force,  and  of  the  dimensions  of 

Fig.  65. 


BODY  PLAN  OP  H.  M.  S.  '  WABRIOB.' 

gun  requisite  to  produce  it,  are  of  perfectly  simple  computation. 
The  punching  pressure  is  about  the  same  as  that  required  to  tear 
asunder  a  bar  of  iron  of  the  same  sectional  area  as  the  surface 
cut  by  the  punch,  which  is  about  60,000  Ibs.  per  square  inch ; 
and  this  pressure  must  act  through  such  a  distance  as  will  suffice 
to  overcome  the  continuity  of  the  metal.  The  distance  through 
which  iron  stretches  before  it  breaks  is  quite  well  known,  and 
this  distance,  multiplied  by  the  separating  pressure  per  square 


FUTILITY    OF    EXISTING   ARMOUR. 


459 


inch,  gives  a  measure  of  the  power  required.  The  velocity  of 
cannon  balls  is  also  known,  and,  by  the  law  of  falling  bodies,  the 
height  from  which  a  body  must  descend  by  gravity  to  acquire 
that  velocity  can  easily  be  determined ;  and  the  weight  of  the 
ball  in  Ibs.,  multiplied  by  this  height  in  feet,  must  always  bo 
greater  than  the  punching  pressure  in  Ibs.  multiplied  by  the  dis- 
tance in  parts  of  a  foot  through  which  iron  stretches  before  it 
breaks,  else  the  ball  will  not  penetrate.  We  have  by  no  means 

Fig.  66. 


TRANSVERSE  SECTION  OF  H.  If.  S.  '  WARUIOR.' 

reached  the  limit  of  the  power  of  projectiles,  nor  is  the  explora- 
tion of  those  limits  yet  begun.  Piston  guns  may  be  made  in 
which  the  projectile  would  consist  of  a  cigar-like  body,  or  thun- 
derbolt, with  spiral  fins  supporting  a  wooden  piston  or  wad, 
which  would  transmit  to  a  projectile  of  small  diameter  the  pow- 
er generated  in  a  cylinder  of  large  diameter.  A  gun  is  virtually 
a  cylinder,  and  the  ball  is  the  piston ;  and  the  power  given  to 
the  ball  will  be  represented  by  the  pressure  exerted  by  the  ex- 


460  STEAM   NAVIGATION. 

ploding  powder  multiplied  by  the  capacity  of  tlie  gun.  As,  how- 
ever, there  are  practical  limits  to  the  length  of  a  gun,  it  may  he 
advisable  to  increase  the  diameter,  in  order  to  get  the  requisite 
power.  But  this  must  be  done  without  increasing  the  diameter 
of  the  ball,  which  would  encounter  greater  resistance  if  made  too 
large;  and  piston  guns  are  the  obvious  resource  in  such  a  case — 
the  piston  being  so  contrived  that  it  would  be  left  behind  by  the 
ball  so  soon  as  it  had  left  the  mouth  of  the  gun,  and  had  acquir- 
ed all  the  power  which  a  piston  could  communicate.  The  pro- 
jectile itself  should  have  a  sustaining  power  as  well  as  a  pro- 
jectile one,  to  which  end  it  should  contain  a  certain  quantity  of 
rocket  composition  that  would  burn  during  the  flight  of  the  ball ; 
and  as  the  velocity  of  the  ball  would  be  high,  the  rocket  gas 
would  operate  with  little  slip,  and  with  much  greater  efficiency, 
therefore,  than  in  rockets.  The  spiral  feathers  would  cause  the 
projectile  to  revolve  in  its  flight,  in  the  same  manner  in  which  a 
patent  log  is  turned  by  the  water ;  and  any  need  for  rifling  the 
gun  would  thus  be  obviated,  as  the  air  would  act  the  part  of  the 
rifle  grooves.  By  these  means  far  greater  ranges  and  far  greater 
accuracy  of  aim  may  be  obtained  than  is  at  present  possible,  and 
it  needs  no  great  perspicacity  to  see  that  the  success  of  maritime 
warfare  will  henceforth  depend  on  the  speed  of  the  vessels  em- 
ployed, and  the  range,  force,  and  accuracy  of  the  projectiles.  A 
small  and  very  swift  steamer  with  projectiles  of  the  kind  I  have 
described  would  be  able  to  destroy  at  her  leisure  a  vessel  like  the 
'  Warrior,'  while  herself  keeping  out  of  range  of  the  best  existing 
guns  which  the  assailed  vessel  could  bring  to  bear  against  her 
opponent.  "With  great  accuracy  of  aim,  and  by  choosing  a  posi- 
tion where  the  wind  would  have  little  disturbing  influence,  a 
large  vessel  could  be  struck  at  a  distance  at  present  deemed  chi- 
merical, and  a  few  of  such  vessels  as  I  have  described,  without 
any  armour  at  all,  would  speedily  disable  any  vessel  which  was 
not  provided  with  the  same  species  of  projectile.  Even  if  the 
large  vessels,  however,  were  to  be  armed  with  projectiles  of 
equal  range  and  power,  the  advantage  would  still  be  with  the 
small  vessels,  as  they  would  be  more  difficult  to  hit ;  and  by 
taking  up  an  external  position  and  firing  their  guns  in  converg- 


FEATURES    OF    AMERICAN    MONITORS.  461 

iiig  lines,  of  which  the  assailed  object  would  be  the  focus,  a  great 
advantage  would  be  given  in  the  attack. 

The  vessels  called  Monitors,  recently  constructed  in  America, 
and  which,  I  believe,  owe  their  most  valuable  features  to  the 
talents  of  Ericcson,  the  eminent  Swedish  engineer — whose  ser- 
vices were  lost  to  this  country  through  the  incapacity  of  the  Ad- 
miralty at  the  time  of  the  introduction  of  the  screw-propeller — 
are  a  very  judicious  embodiment  of  the  leading  principles  of  iron- 
clad vessels  so  as  to  secure  the  greatest  possible  efficiency.  The 
constructors  of  those  vessels  saw  that  the  thickness  of  the  sides 
must  be  very  much  greater  than  it  is  in  our  iron-clads,  to  pre- 
vent heavy  shot  from  going  through  them ;  and  this  thickness  is 
reconciled  with  the  usual  buoyancy  by  making  the  sides  of  the 
vessel  very  low,  so  that  only  a  small  area  has  to  be  protected. 
Very  powerful  guns  are  employed  in  these  vessels;  and  as  it 
would  be  difficult  to  manoeuvre  such  guns  by  hand,  a  steam-en- 
gine is  introduced  for  this  purpose,  which  gives  great  facility  in 
the  handling.  To  protect  the  guns  and  gunners  from  hostile 
shot,  they  are  placed  in  towers  of  iron,  the  metal  of  which  is  15 
inches  thick,  and  these  towers  are  turned  like  a  swing-bridge  to 
enable  the  gun  to  be  pointed ;  but  the  mechanism  is  so  contriv- 
ed, that  the  hand  of  a  child  acting  on  the  engine  will  suffice  to 
move  the  tower.  Admiral  Porter  states  that  a  Monitor  of  this 
construction  would  be  able  to  cross  the  Atlantic,  and  attack  and 
sink  our  iron-clads  at  her  leisure,  without  being  herself  liable  to 
injury ;  and  I  think  he  is  right  in  his  conclusion,  though  it  was 
a  most  indelicate  thing  for  him  to  have  indicated  such  an  occu- 
pation for  this  class  of  vessels.  But  persons  who  infer  the  help- 
lessness of  this  country  to  resist  such  attacks,  from  .the  imbecili- 
ty of  the  Admiralty,  will  find  themselves  mistaken ;  and  there 
are  obviously  two  ways  in  which  such  Monitors  could  be  de- 
stroyed. Those  vessels,  though  immensely  strong  above  the 
water,  are  weak  below,  being  there  without  armour,  as  they  are 
protected  from  shot  by  the  water.  But  a  vessel  like  the  '  War- 
rior,' if  armed  in  a  line  with  the  keel — or  a  little  above  it — with 
a  great  steel  blade  or  horn  40  or  50  feet  long,  would  by  running 
against  a  Monitor,  break  into  the  bottom  and  sink  her.  Such  a 


462  STEAM   NAVIGATION. 

conflict  would  be  like  a  sword-fish  attacking  a  whale ;  and  the 
horn  or  blade  would  in  no  way  affect  the  steering  of  the  vessel, 
as  it  would  only  virtually  make  her  so  much  longer.  Another 
way  in  which  Monitors  could  be  destroyed,  is  by  running  over 
them.  As  they  are  not  many  feet  out  of  the  water,  to  submerge 
them  for  a  few  feet  more,  by  placing  a  corresponding  weight 
upon  their  deck,  would  sink  them  altogether ;  and  if  we  suppose 
a  vessel  with  a  very  raking  stem,  and  so  trimmed  by  the  stern  as 
to  bring  the  forefoot  out  of  the  water,  to  be  run  against  a  Moni- 
tor, it  will  be  obvious,  if  the  vessel  be  a  large  and  heavy  one,  and 
the  speed  of  propulsion  be  high,  that  she  would  run  up  on  the 
deck  of  the  Monitor,  and  sink  her  at  once.  The  weight  and 
speed  of  vessel  that  would  work  this  catastrophe  in  the  case  of 
any  given  Monitor,  is  matter  of  simple  calculation ;  and  it  is 
quite  an  error,  therefore,  to  imagine  that  any  Monitor  yet  con- 
structed might  not  be  promptly  disposed  of.  Certainly  they 
might  be  made  tight,  like  diving-bells,  so  that  even  if  sunk  and 
ridden  over,  they  would  come  up  again.  But  this  would  be  a 
difficult  thing  to  do ;  and  even  if  it  were  done,  the  next  step 
would  be,  that  the  attacking  vessel  would  not  go  over,  but  would 
stop  upon  them.  No  doubt  the  Monitor  might  as  easily  run 
into  the  attacking  vessel  as  the  attacking  vessel  into  her,  pro- 
vided the  Monitor  had  equal  speed.  But  the  construction  of 
Monitors  is  not  favourable  for  speed ;  and  if  speed  is  to  settle 
the  question,  there  is  no  need  for  iron  plating.  The  fact  is,  such 
infallible  recipes  for  victory  as  Monitors  are  supposed  to  consti- 
tute, almost  always  break  down.  I  believe  that  such  vessels 
may  be  made  sea-worthy ;  they  may  be  made  impenetrable  to 
any  guns  at  present  in  our  navy,  and  the  guns  they  mount  may 
be  able  to  riddle  our  iron-clads  like  so  many  ships  of  card-board. 
All  that  I  grant.  But  guns  can  be  made  to  go  through  the  tow- 
ers and  sides  of  Monitors,  though  twice  as  thick  as  they  are ;  all 
the  existing  Monitors  can  easily  be  outstripped  in  speed ;  and 
vessels  with  steel  horns  may  rip  up  their  bottoms,  and  ves- 
sels built  with  greatly  slanted  stems  may  be  made  to  run  over 
and  sink  them.  It  is  true  there  are  the  guns  of  the  Monitor  to 
be  encountered  by  the  attacking  vessel.  But  if  that  vessel  has 


FEATURES    OF    AMERICAN    MONITORS. 


4G3 


464  STEAM    NAVIGATION. 

several  decks,  and  if  the  deck  over  the  main  hold  be  made  into 
a  water-tank,  with  water-tight  trunks  communicating  between 
the  hold  and  the  decks  above,  a  shot  between  wind  and  water 
would  not  let  water  in,  as  the  space  is  filled  witli  water  already; 
and  the  attacking  vessel,  therefore,  could  not  be  sunk  by  any  fire 
the  Monitor  could  bring  against  her,  unless  it  could  be  made  to 
pierce  through  the  sea  so  as  to  enter  the  lower  hold  by  which 
the  flotation  is  given.  With  the  low  elevation  of  the  Monitor 
turrets,  however,  this  does  not  appear  to  be  a  probable  contin- 
gency. Small  rocket-vessels,  propelled  at  a  high  speed  by  rock- 
et gas  issuing  at  the  stern  beneath  the  water,  will  probably  be 
used  in  actual  warfare  for  many  purposes ;  and  the  same  resource 
may  be  employed  temporarily  to  increase  the  speed  of  large 
steamers.  If,  for  example,  the  iron-clads  of  the  '  "Warrior '  type 
had  a  tube  opening  beneath  the  water  at  each  quarter,  out  of 
which  rocket  flame  and  gas  were  made  to  issue,  the  speed  of  the 
vessel,  while  the  emission  lasted,  would  be  increased ;  and  this 
temporary  acceleration  might  suffice  to  give  her  a  decisive  supe- 
riority over  an  opponent. 


INDEX. 


ABS 

A  BSOLUTE  zero,  187 
£.    Addition,  nature  of,  10;   addition 
table,  11 ;  method  of  performing,  11 ; 
examples  of,  12 

—  of  fractions,  84 

—  indicated  by  +  or  pins,  10 
Air,  composition  of,  174 

—  dilatation  of,  by  heat,  145, 147 

—  height  of  column  of,  to  produce  atmos- 
pheric pressure,  100 ;  relative  density 
of,  101 

—  into  a  vacuum,  velocity  of,  101 

—  in  water  lowers  the  boiling  point,  163 

—  pump  and  condenser,  proportions  of, 
215,  222 

Indicator  diagrams  taken  from,  844, 

845,  351,  857 

studs  in  side  levers,  269 

of  marine  engines,  proper  propor- 
tions of,  280 

rod  of  marine  engines,  proper  pro- 

tlons  of,  280 ;  table  of  proportions  of, 
299 

side  rods  of  marine  engines,  284 

rod  of  land  engines,  232 

crosshead,  proper  dimensions  of, 

282 

bucket,  cutter  of.  281 

Algebra,  wherein  It  differs  from  arithme- 
tic, 10 

Allen's  engine,  diagrams  from,  357 

American  monitors,  461 

Annular  valves,  how  to  compute  the 
pressure  on,  212 

AppoM's  centrifugal  pump,  886 

Archimedes  screw,  886 

Arithmetic  of  the  steam-engine,  1 

—  defined,  5 

—  wherein  it  differs  from  algebra,  10 


BOI 

Armour  of  ships  of  war  penetrable,  458; 
measure  of  its  resistance  to  shot,  459 

Atmospheric  pressure,  height  of  column 
of  air  required  to  produce,  101 

Attraction  of  gravity,  93 

Augmented  surface  of  a  vessel  a  measure 
of  resistance,  443 

Auxiliary  propulsion  of  common  steam- 
ers by  rockets,  464 

•RACK  links,  282 

JJ    Barley  mill,  887 

Barlow's  experiments  on  the  strength  of 

woods,  127 

'  Barossa,'  diagram  from,  350 
Battering  ram,  momentum  of,  105 
Beam  of  land  engines,  233 
Beams,  how  to  determine  strength  of,  87 

—  cast-iron,     Hodgkinson's     rule     for 
strength  of,  138 

Bean  mill,  888 

Bearings,  friction  of,  how  to  limit,  121 ; 

variations  of  velocity  and   pressure, 

122 
Blast  orifice  in  locomotives,  area  of,  313 

—  pipe  In  locomotives,  831 

Block  and  tackle,  weights  movable  by, 

82 

Bochet's  experiments  on  friction,  119 
Bodies,  falling,  laws  of,  93,  97 

—  revolving,   centrifugal  force  of,  109; 
bursting  velocity,  110 

Body-plan  of  steamer  'Fairy,'  454;  of 
4  Rattler '  455 ; '  Bremen,1 456 ; 4  Persia,' 
457;  '  Warrior,' 45S 

Boilers,  circulation  of  water  Jn,  very  Im- 
portant, 173 

—  proportions  of,  308 


466 


INDEX. 


Boilers,  power  measurable  by  evapora- 
tion, 809 

—  wagon,  proportions  of,  310;  Hue,  311 

—  haystack,  by  D.  Napier,  31G ;  by  Earl 
of  Dundonald,  316 

—  strength  of,  820,  322 

—  stays,  321 

—  cylindrical,  proper  diameter  for  given 
pressure  of  steam  and  thickness  of 
plate,  324 ;  safe  pressure  in,  825 

—  bursting  and  safe  working  pressures 
of,  326 

—  of  modern  construction,  heating  sur- 
face of,  375 

—  uptake,  sectional  area  of,  376 

—  and  surface  condensers,  relative  sur- 
face areas  of,  380,  382 

—  fed  by  Gifl'ard's  injector,  3S3 

—  marine,  bulk  of,  313 

—  locomotive,  example  of,  329 
Boiling  point  of  water  raised  by  molec- 
ular attraction,  168 

—  lowered  by  presence  of  extraneous 
substances,  168 

Boulton  and  Watt's  rule  for  the  fly- 
wheel, 228 

system  in  drawing  office,  289 

Bourne's  duty  meter,  3T3 

Boutigny,  his  experiments  on  spheroidal 
condition  of  liquids,  169 

Breadth,  maximum,  of  ships,  best  posi- 
tion of,  417 

'  Bremen,'  steamer,  456 

Brule,  membrane  pump  by,  885 

Bucket  of  air-pump,  cutter  through, 
281 

Bulk  of  marine  tubular  boilers,  313 

Bursting  and  safe  working  pressures  in 
boilers,  326 

—  velocity  of  fly-wheels,  110 


pAIRD  and  Co.,  dimensions  of  side  le- 
\J    ver  marine  engines  by,  287 

engines  of '  Hansa '  by,  814 

Canals,  velocity  of  water  in,  165,  433 
Cannon  ball,  momentum  of,  105 
Carbonic  acid,  specific  gravity  of,  175 

—  oxide  produced  in  bad  furnaces,  176 
Carl-Metz,  pumps  by,  885 

Cast-iron,  limit  of  load  on  in  machinery, 

88 

beams,  proportions  of,  88 

columns,  strength  of,  131 ;  beams, 

133 

Centigrade  thermometer,  188 
Centres  of  gyration  and  percussion,  112 
oscillation,  115 

—  in  side  lever,  269 

Centrifugal  force,  10T;  how  to  determine 

the,  108;  bursting  velocity,  110 
Centrifugal  pendulum  or  Governor,  117 

—  pum] 


CON 

Characteristic  in  logarithms,  nature  of,  55 
Cheapest  source  of  rower,  131 
Chelsea  Water  Works,  engines  of,  36S 
Chimneys,  exhaustion  produced  by,  304, 
Jioulton  and  Watt's  rule  for  proportion 
of,  in  land  boilers,  305,  813 ;  in  marine 
boilers,  305,  314;  1'eclet's  rule  for  pro- 
portions of,  306 

—  proper  height  of.  305 

—  sectional  area  of,  required  to  evaporate 
a  cubic  foot  per  hour,  318 

Circular  and  square  inches,  8 
Circulation  of  water  in  bodies  very  im- 
portant, 173 

heat,  171 

Circular  saw,  389 

—  loom,  398 

Circumscribing  parallelepiped,  406 
'Clyde'  steamer,  dimensions  of,  287 
Coals,  heating  powers  of  different,  177 

—  consumed  per  square  foot  of  fire  bars 
to  evaporate  a  cubic  foot  per  hour,  814, 
816 

indicated  horse  power  per  hour, 

at  Chelsea  Water  Works,  368 
Coefficient  multiplier,  77 

—  of  friction,  119 

Coefficients  of  various  steamers,  77 

—  of  dilatation  of  gases,  144 
Cohesion  of  water,  168 

Coke  burned  in  locomotives,  381 

Collapsing  pressure  of  flues,  827 

Colours,  how  produced,  94 

Columns,  laws  of  strength  of,  128, 181 

Cold  water  pump,  to  find  the  proper  ca- 
pacity of,  227 

Combustion,  nature  of,  174;  air  required 
for,  174;  total  heat  of,  176;  rates  of, 
179 

Combustibles,  evaporative  powers  of, 
175 

Common  divisor  defined,  33 

—  denominator,  how  to  reduce  fractions 
to,  35 

Compound  quantities,  57 
Compressibility  of  gases,  148 
Conical  measure,  9 
Condensation  of  steam  by  cold  surfaces, 

173 ;  secret  of  refrigerative  efficiency. 

178 
Condenser  and  air-pump,  proportions  of, 

215 

—  and  boiler  surfaces,  proportionate  areas 
of,  173 

Condensers,  proper  construction  of,  815 

—  surface,  cause  internal  corrosion   in 
boilers,  3S1 ;  proportions  of,  in  recent 
cases,  382 

Conduction  of  heat,  172 
Conducting  powers  of  metals,  172 
Conservation  of  energy,  78 

—  force,  78 

Connecting  rod  of  land  engines,  288 


IXDEX. 


467 


CON 

Connecting  rod  of   marine  engines  of 

wrought-iron,  263 
Consumption  of  fuel  at   Chelsea  Water 

Works,  363 

—  of  coal  in  steamer  'Hansa,'  315 
Cooling  surface  of  condensers,  315 
Corrosion  of  boilers  internally  with  sur- 
face condensers,  3S1 

Cotton  spinning  mill,  391 
Counter,  872 

Cranes,  weights  lifted  by,  82 
Crank,  strain  from  infinite,  90 
Crank,  shafts  of  cast-iron,  Mr.  "Watt's  rule 
for,  239 

—  large  eye  of,  when  cast-iron,  240 

—  when  of  cast-iron,  242 

—  table  of  proportions  of,  296,  29T,  298, 
300 

—  pin,  when  of  cast-iron,  246 
journal,  271 

of  marine  engines,  table  of  propor- 
tions of,  300 

Cranks  for  marine  engines  of  wrought- 
iron,  271,  278 

Crosshead  of  marine  engines,  proper  pro- 
portions of,  255 

depth  of,  rule  for,  256 

—  eye  of,  257 

—  of  air  pump,  proper  dimensions  of, 
282 

Crosstail,  proper  proportions  of,  267 
Cross  section  of  ships,  best  form  of,  409 
Crucible,  red  hot,  ice  made  in,  170 
Cube  roots,  nature  of,  43 
of  fractions,  48 

—  root,  method  of  extracting,  50 
Cubes  and  cube  roots,  48 
Cubic  measure  explained,  76 
Cushioning,  diagrams  showing,  855,  356 
Cutting  off  the  steam,  advantage  of,  182 
Cutter  through  piston,  263 

air-pump  bucket,  281 

—  of  connecting  rod,  265 

Cutters  and  gibs.    See  GIBS  and  CET- 

TEBS. 
Curves,  mode  of  representing  dimensions 

by,  288 
Cylindrical  measure,  9 


DAECT9  experiments  on  the  friction 
of  various  surfaces  in  water,  439 
Decimal  system  of  numeration,  2 
Decimal  fractions,  nature  of,  6 
Denominator  of  fractions  defined,  6 
—  common,  how  to  reduce  fractions  to, 

85 
Density  of  water,  maximum,  139 

steam  of  atmospheric  pressure,  102 

Densities  of  gases,  166 
Diagrams,  indicator,  how  to  read.  887 ; 
how  to  take,  870;  various  examples  of, 
888 ;  from  air-pump,  844, 851, 857 ;  from 


EVA 

hot  well,  360,  362 ;  from  water  pump, 
361 ;  double  cylinder,  869  —  showing 
momentum  of  indicator  piston,  847 

Diameter  of  cylindrical  boiler  proper  for 
given  pressure  and  thickness  of  plate, 
325 

'Dictator'  steam  ram  or  monitor,  con- 
structed in  America  by  Ericsson,  463 

Differential  motions  for  raising  weights, 
85 

—  gearing,  66 

Dilatation,  140:   force  of,  148;  of  gases, 

144,145 
Dimensions  of  engines  laid   down   to 

curves,  288 
side   lever  marine   engines,   254, 

801 
marine  engines  by  Caird,  287 ;  by 

Maudslay,  290 ;  by  Se.ward,  292 

locomotive  engines,  801 

Disc,  revolving  power  resident  in,  111 
Divisor  defined,  24 

—  common,  defined,  33 
Dividend  defined,  24 

Division,  nature  of,  24;  examples  of,  26; 
explanation  of,  29 

—  of  fractions,  88 

Donny,  his  experiments  upon  ebullition, 
168 

Double  cylinder  engines,  diagrams  from, 
36-2,  365,  869 

Drawing  of  one  engine  suitable  for 
another  by  altering  scale,  287 ;  conven- 
ient sizes  for  drawings,  2S9 

Duke  of  Sutherland's  yacht,  diagrams 
from,  257,  260 

Dundonald,  Earl  of,  boilers  by,  816 

Duty  of  engines  at  Lambeth  Water- 
works, 868 

Duty  meter,  378 

Dynamical  unit,  79 

Dynamometer,  872 


,  168 
Hi    Elastic  force  of  steam  at  different 

temi>erfttures,  159 
Elbow-jointed  lever,  89 
Elliot,  Brothers,  indicator  by,  835 
Energy,  conservation  of,  78 
Engines,  if  perfect,  power  producible  by, 

181 

Equations,  nature  of,  T4 
Equation  for  determining  the  speed  of 

steamers,  76 

Equivalent,  mechanical,  of  heat,  91 
Ericsson  the  designer  of  the  American 

monitors,  461 
Evaporation,  latent  heat  of,  162 

—  in  locomotives,  331 

Evaporative    powers   of    combustibles, 
175 

—  power  of  coal,  812 


468 


INDEX. 


EXH 

Exhaustion  of  chimneys,  304 
Expansion  of  air  by  heat,  146 

—  of  gases,  166 

—  by  link,  diagrams  showing,  355,  356 

—  of  steam,  182;  measure  of  benefit  from, 
183 ;  mean  pressure  of  expanding  steam, 

loO 

—  producible  by  a  given  proportion  of 
lap  to  stroke  of  valve,  1ST,  188 

Expansion  producible  by  throttling  the 

steam,  198 

Exponents,  fractional,  51 
Eyes  of  cranks  of  wrought-iron,  271,  272 
Eye  of  air-puinp  crosshead",  283 


TUCTOES  denned,  29 
J?     Fahrenheit's  thermometer,  133 
'Fairy'  steamer,  lines  of,  454 
Falling  bodies,  laws  of,  90,  97 
Fans,  power  required  to  drive,  391 
Feed  pipe,  rule  for  proportioning,  221 

—  pump,  to  find  the  proper  capacity  of, 
224 

Feeding   boilers   by  Giffard's   injector, 

384 

Film  of  water  moving  with  a  ship,  428 
Fire  bars  of  locomotives,  314 
Fishes,  shape  oij  translatable  into  that  of 

ships,  407 

Flaud,  pumps  by,  385 
Flax  mills,  395 

Floating  bridge,  diagrams  from,  855,  356 
Flour  mill,  387 
Flues,  proper  sectional  area  of,  310,  811 

—  boilers,  proper  proportions  of,  811 
Flues,    sectional    area    of,   required   to 

evaporate  a  cubic  foot  of  water  per 
hour,  314 

—  collapsing  pressure  of,  32T 
Fluids,  motion  of,  100 

Fly-wheels,  momentum  of,  106  ;  burst- 
ing velocity  of,  110 

—  should  have  power  equal  to  six  half 
strokes,  216 

Fly-wheel,  Boulton  and  Watt's  rule  for 
the,  228 

—  shaft,  239,  240 

Force,  conservation  of,  78 

—  centrifugal,  107 ;  how  to  measure,  108 ; 
bursting  velocity,  110 

—  of  dilatation,  143 

—  elastic,  of  steam  at  different  temper- 
atures, by  M.  Kegnault,  159 

Form  of  least  resistance  in  ships,  402 

Formula  for  determining  the  speed  of 
steamers,  76 

Foot  valve,  passages  to  find  the  proper 
area  of,  228 

Fractions,  nature  of,  5 ;  vulgar,  5 ;  deci- 
mal, 6 

—  multiplication  by,  9 

—  nature  and  properties  of,  31 ;  how  to 


GRA 

reduce  a  fraction  to  its  lowest  terms. 
33 

—  addition  and  subtraction  of,  34 

—  how  to  reduce  a  common  denominator, 
35 

—  multiplication  and  division  of,  38 

—  squares  and  square  roots  of,  45 

—  cubes  and  cube  roots  of,  48 

—  resolvable  into  infinite  series,  66 
Fractional  exponents,  52 

Frame,  midship,  of  ships,  best  position 
of,  417 

Franklin  Institute,  experiments  on 
steam  by,  157 

French  Academy,  experiments  on  steam 
by,  157 

Friction,  118;  coefficient  of,  119;  experi- 
ments on,  by  Morin  and  Bochet,  118 

—  of  crank  pins,  120 

bearings  varies  with  the  pressure, 

121 ;  relations  of  pressure  and  velocity, 
122 

Friction  of  flowing  water,  199 

engines,  367 

water  in  pipes  does  not  vary  with 

the  pressure,  429 

bodies  moving  in  water  varies  with 

nature  of  surface,  439 

the  bottom  the  main  source  of  re- 
sistance in  ships,  423 

bottom  of  steamer,  '  Leinster,'  438 

Fuel,  different  kinds  of,  heating  power, 
175 

—  consumed  per  indicated  horse  powei 
per  hour  at  Chelsea  Water-works,  868 

Fulling  mills,  394 
Furnaces,  temperatures  of,  178 
rates  of  combustion,  179 

—  importance  of  high  temperature  of,  377 

GAS  into  a  vacuum,  velocity  of,  102 
Gases,  dilatation  and  compression  of, 
143, 144, 145 

—  and  vapours,  difference  between,  158 

—  liquefied  by  cold  and  pressure,  153 

—  specific  heats  of,  164,  165;  densities, 
volumes,  and  rates  of  expansions  of, 
166 

Gearing,  differential,  86 

Gearing,  proportions  proper  for,  231 

Gibs  and  cutters  of  crosshead,  258 

side  rods.  260 

through  crosstail,  266 

air-pump  crosshead,  280 

air-pump  side  rods,  286 

Giffard's  injector,  383 

Glass  works,  897 

Governor  for  steam  engines,  117 

—  to  determine  the  right  proportions  of 
the,  281 

Grate  coal  burned  on  each  square  foot  in 
different  boilers,  314 


INDEX. 


469 


GEA 

Grate  surface  to  evaporate  a  cubic  foot 
per  hour,  314 

—  bars    per   nominal   horse-power    in 
steamers,  315 

Gravity,  nature  of,  93 
Gudgeons  in  side  lever,  269 
Guns,  piston,  460 

Gyration ,  centre  of,  112 ;  to  find  the  posi- 
tion of,  113 

Gyroscope,  phenomena  of  the,  93 
Gwynn's  centrifugal  pump,  386 

'TTAJfSA'    steamer,    proportions    of 
11    machinery  of,  314 
Haystack  boiler,  316 
Heat,  motive  power  of,  90 

—  mechanical  equivalent  of,  91, 167 
power  producible  by,  131 

—  sensible,  defined,  135 

—  latent,  defined,  135 

—  specific,  defined,  135 

—  dilatation  by,  140 

—  specific,  162 

—  unit  of,  162 

—  effect  of,  in  accelerating  the  velocity 
of  rivers,  425 

Heating  surface  of  boiler  per  square  foot 
of  fire  grate,  314 

to  evaporate  a  cubic  foot  of 

water  per  hour,  314 

and  cooling  surface  of  con- 
denser, 815 

in  modern  boilers,  375 

Height  from  which  bodies  have  fallen 
determinable  from  their  velocity,  98 

Height  from  which  bodies  have  fallen 
determinable  from  their  time  of  fall- 
ing, 98 

—  of  chimney  proper  for  different  boilers, 
805 

Hodgkinson,  strength  of  woods  accord- 
ing to,  128 ;  law  of  strength  of  pillars 
by,  128, 131 ;  of  cast-iron  beams,  183 

Horse-power,  nominal,  definition  of,  79 

actual,  definition  of,  79 

Hot  well,  Indicator  diagrams  from,  359, 
860 

Hydraulic  press,  pressure  producible  by, 
81,84 

—  head  of  water  different  from  hydro- 
static head,  426 

—  mean  depth  of  a  ship,  481 
Hydrostatic  resistance  of  vessels  increas- 
es with  speed  and  with  breadth,  422 

—  head  of  water  different  from  hydraulic 
head,  426 


ICE,  weight  of  at  82%  140 
—  made  in  a  red-hot  crucible,  170 
Improvements  required  in  boilers  and 
condensers,  879 


LES 

Inches,  square  and  circular,  spherical, 

cylindrical,  and  conical,  9 
Incommensurables,  nature  of,  46 
Indian  system  of  numeration,  3 
Indicator,    construction    of    the,    833; 

Kichards',  334 ;  method  of  applying  the, 

335,870 

—  diagrams,  how  to  read  835  ;  how  to 
take,  370 ;  various  examples  of,  838 ; 
from  air-pump,  344,  351,  357  ;    from 
hot  well,  359 ;  from  water  pump,  361 ; 
from    double    cylinder    engine,   365, 
869 

Indicator  diagrams,  showing  momentum 

of  indicator  piston,  347 
Inertia  defined,  105 
Infinite  series,  how  to  resolve  fractions 

into,  66 

—  strains  from  crank  and  elbow-jointed 
lever,  89 

Injection  pipe,  to  find  the  proper  area  of, 

222 

Injector,  Giffard's,  551 
Invisible  light,  96 
Iron,  steel,  and  other  metals,  strength 

of,  125 

—  fusible  at  low  temperatures,  151 

—  works,  897 

Iron-clad  steamers  penetrable  by  shot, 

457 

Irrational  numbers  defined,  46 
1  Island  Queen,1  indicator  diagram  from, 


JET,  composite,  in  chimney,  819 

tl  Joule's  experiments  on  the  conden- 
sation of  steam,  173 

Journals  of  crosshead,  proper  dimensions 
of,  257 

—  air-pump  crosshead,  284 


T  AMBETH  "Water-works,  engines  at, 
JL    862;  diagrams  from,  865;  duty  of, 

868 
Latent  heat  defined,  135 

of  liquefaction,  151 

—  heats  of  steams  from  water,  alcohol, 

ether,  and  sulphuret  of  carbon,  154 
Lap  of  valve  proper  for  a  given  amount 

of  expansion,  187, 189. 190 
on  eduction  side,  effects  of.  187, 

193 

Lead  plug,  830 

'Leinster,'  steamer,  computation  of  fric- 
tion of,  438 
Length  of  pendulum  to  vibrate  at  any 

given  speed,  115 
vessels  should  vary  with  intended 

speed,  421 
Leslie's  explanation  of  the  strength  of 

iron,  126 


470 


LET 

Lctcstu,  pumps  by,  385 
Lever,  action  of  the,  83 

—  elbow-jointed,  89 
Levers  of  Stanhope  press,  89 
Light,  invisible,  94 

Lineal  measure  explained,  7 

Lines  of  ships,  400 ;  illustrated  by  shape 

of  fishes,  40T 
Link  motion,  198 

—  expansion  by,  diagrams  showing,  355, 
356 

Liquefaction,  150 ;  latent  heat  of,  151 ;  of 

gases,  153 
Liquids,  dilatation  of,  by  beat,  143 

—  specific  heats  of,  164 
Locomotive  engines,  proper  proportions 

of,  301 

—  boiler,  example  of,  329 

—  efficiency  of  steam  vessels,  814 
Logarithms,  nature  of,  52 ;  mode  of  using, 

56 

Lowest  terms,  how  to  reduce  a  fraction 
to,  33 


TITADAGASCAE,  mode  of  numeration 

M    used  in,  2 

Machines,  strains  and  strengths  of,  81,  87 

—  how  to  determine  power  of,  82 
Magnitude,  standards  of,  7 

Magnus,  his  experiments  upon  ebulli- 
tion, 168 
Main  links,  232 

—  centre  of  land  engines,  232 

—  centre  of  marine  engines,  268 

—  beam  of  land  engines,  how  to  propor- 
tion, 233 

Maize  mill,  3S7 

Marine  engines,  proportions  of  the  parts 
of,  254-301 

—  boilers,  proportions  of,  314 
Marquis  do  I'lftipital,  his  rule  for  finding 

the  centrifugal  force,  108 
Materials,  strength  of,  124 
Maudslay  and  Co.'s  side  lever  engines, 

dimensions  of,  290 
Maximum  density  of  water,  139 
Midship  section  of  ships,  best  form  of, 

409 

—  frame  of  ships,  best  position  of,  417 
Mill  gearing,  proportions  proper  for,  246 
Mills:  flour,  387;  barley,  387;  rye,  387; 

maize,  887 ;  bean,  388 ;  oil,  388 ;  saw, 
888 ;  sugar,  390 ;  cotton,  391 ;  weaving, 
393 ;  wool,  393 ;  fueling,  394 ;  flax,  394 ; 
paper,  396 ;  rolling,  397 

Millwall  Ironworks,  engines  by,  882 

Mechanical  power  from  the  Bun,  79 

nature  of,  90 

of  the  universe  constant,  92 

—  equivalent  of  heat,  91 
Melting  points  of  solids,  148 
Membrane  pump,  by  Brule, 


PER 

Mercury,  relative  density  of,  100 

—  into  a  vacuum,  velocity  of,  101 
Merry  weather,  pumps  by,  385 
Metals,  strengths  of,  125 

—  conducting  powers  of,  172 
Molecular  attraction  of    water  retards 

boiling,  168 

Momentum  defined,  105 ;  of  rams,  105 ; 
of  cannon  balls,  105 

—  of  heavy  moving  bodies,  how  meas- 
ured, 106;  of  a  revolving  disc,  111 

indicator  piston,  347 

Monitors,  features  of  their  construction, 
461 ;  weak  points  of,  462 ;  mode  of  de- 
stroying, 463 

Moors  brought  decimal  system  into  Eu- 
rope, 3 

Morin's  experiments  on  friction,  118 

Morin,  General,  his  experiments  on  va- 
rious machines,  387-397 

Motive  power  of  heat,  90 

Motion  of  fluids,  100 

—  power  required  to  produce,  106 

—  in  a  circle,  107 

Multiplication,  nature  of,  16;  multipli- 
cation table,  19,  23  ;  examples  of,  20 ; 
mode  of  performing,  22 

Multiplier  defined,  20 

Multiplicand  defined,  20 

Multiplication  by  fractions,  9 

—  of  fractions,  88 

'  Munster,'  indicator  diagrams  from,  340 
Mylne,  his  constant  for  velocity  of  water 
in  pipes,  206 


ATAPIEE,  DAVID,  his  haystack  boil- 

ll     ers,  173,  316 

Numerator  of  fractions  defined,  5 

OAK  posts,  proper  load  for,  180 
Oil  mill,  888 
Ordnance,  increased  power  of,  attainable, 

459 
'  Orontcs'  indicator,  diagrams  from,  348, 

349 

Oscillation,  centre  of,  114 
Oxygen  required  for  combustion,  175 


207 


T)ADDLE  shaft,  294 
I      Paper  mill,  896 
Parallel  motion,  how  to  describe  the, 
Parallelepiped,  circumscribing,  406 
Peclet's  rule  for  proportions  of  chim- 
neys, 306 
Pendulum,  action  of  the,  95 

—  laws  of  the,  114 

—  centrifugal,  116 
Percussion,  centre  of,  112 
Perrin,  pumps  by,  387 
Perry,  pumps  by,  887 


471 


PEE 

Persian  wheel,  386 
'  Persia,'  steamer,  8S9 
Phipps.  on  resistances  of  bodies  by,  436 
Pillars,  law  of  strength  of,  131 
Pipes,  velocity  of  water  flowing  in,  199, 
433 

—  and  passages,  proper  proportions  of, 
for  different  powers,  300 

Piston  rod  for  land  engines.  231 

of  marine  engines,  261 

table  of  proportions  of, 

299 

—  valves,  by  D.  Thomson,  363 

—  guns,  409 

Plates  of  boilers,  proper  thickness  of, 

322 

Plus,  the  sign  of  addition,  10 
Pneumatic  Despatch  Company's  engine, 

indicator  diagrams  from.  354,  355 
Portsmouth    floating    bridge,  diagrams 

from,  355,  356 

Posts  of  oak,  proper  load  for,  130 
Powers  and  roots  of  numbers,  49 
Power,  m  chanical,  from  the  sun,  79 

—  mechanical,  nature  of,  90 

—  motive,  of  heat,  90 

—  required  to  produce  motion,  106 

—  resident  in  a  revolving  disc,  111 

—  producible  by  a  given  quantity  of 
heat,  181 

in  a  perfect  engine,  181 

—  cheapest  source  of,  181 

—  nominal,  how  to  determine,  208;  Ad- 
miralty rule  for,  211 

of  boilers  an  indefinite  expression, 

809 

—  and  performance  of  engines,  333 

—  loom  weaving,  393 

—  required  to  produce  a  given  speed  in 
steam  vessels,  430,  432,  443 

Press,  Stanhope,  89 

Pressure,  atmospheric,  how  produced, 
100 

—  permissible  on  bearings  moving  with 
a  given  speed,  121 

—  strength  of  boiler  to  withstand,  823 

—  safe,  in  a  cylindrical  boiler,  325,  326 

—  collapsing  of  flues,  327 
Pressures  and  volumes  of  gas,  147 
Printing  machines,  396 
Product  defined,  20 

Projectiles  should  contain  rocket  com- 
position, 460 ;  and  have  spiral  feathers 
to  put  them  into  revolution,  460 

Proportion,  nature  of,  42 

Proportions  of  steam-engines,  208,  214 

engines  laid  down  to  curves,  288 

locomotive  engines,  801 

boilers,  304 

wagon  boilers,  310 ;  of  flue  boilers, 

811 

Pump,  combined  plunger  and  bucket, 


SAW 

Pumps,  relative   efficiency  of  different 
kinds,  337 

—  by  various  makers,  3S7 

Pumping  engine  at  St.  Katherine's  docks, 
diagram  from,  346 

—  engines,  friction    of,    867 ;    duty  of, 
368 


AUOTIEXT  defined,  24 


pADIATIOX  of  heat,  171 

It  Eankine.  his  method  of  computing 
speed  of  steam  vessels,  443 

Ratio,  or  Proportion,  nature  of,  42 

Reaumur's  thermometer,  138 

Red-hot  crucible,  ice  made  in.  170 

Reduction.  67 

Rcgnault,  his  experiments  on  dilatation 
of  gases,  145 

Regnault's  formulae  for  the  elastic  force 
of  steam,  158 

Relative  bulks  of  water  and  steam  at  at- 
mospheric pressure,  102 

Rennie,  tensile  strength  of  metals  ac- 
cording to.  126 

'  Research,'  indicator  diagram  from, 
349 

Resistance  of  vessels,  399 

mainly  caused  by  friction,  423, 

442 

at  bow  and  stern,  436 

—  hydrostatic,  of  vessels,  increases  with 
speed  and  with  breadth,  422 

'  Rhone'  steamer,  proportions  of  engines 

and  boilers  of.  382 
Richards'  Indicator,  334 
Rivers,  velocity  of,  199 

—  have  water  highest  where  stream  is 
fastest,  424;  effect  of  temperature  on 
velocity,  425 

Riveted  joints,  best  proportions  of,  820 ; 

strength  of,  320 
Revolving  bodies,  centrifugal  force  of; 

109 ;  bursting  velocity,  110 
Rocket  vessels  propelled  by  rockets,  a 

new  expedient  of  warfare,  464 
Roman  method  of  numeration,  3 
Roots,  square,  44 ;  cube,  48 
Ropes  tightened  by  pulling  sideways, 

84 

Rule  of  three,  42 
Rye  mill,  387 


OAFETT  valves,  rule  for  proportion- 

ij    ing,  219 

Saw  mill,  388 ;  for  veneers,  889 


472 


INDEX. 


SAW 
Saw  circular,  389 

—  for  stones,  889 

Screw,  pressure  producible  by,  81 

—  differential,  pressure  producible  by, 
81,85 

—  of  Archimedes,  380 

'  Scud,'  diagram  from  hot  well  of,  360 

Seaward  and  Co.'s  side  lever  engines,  di- 
mensions of,  292 

Sectional  area  of  boiler  flues  or  tubes, 
314 ;  of  chimney,  314 

Sensible  heat  defined,  135 

Side  lever,  proper  proportions  of;  267; 
studs  of,  269 ;  thickness  of  eye  round, 
271 

engines,  dimensions  of,  by  Caird 

and  Co.,  2S7 ;  by  Maudslay,  290 ;  by 
Seaward,  292 

—  rods  of  marine  engines,  proper  pro- 
portions of,  258 

—  rods  of  air-pump  in  marine  engines, 
284 

Solid  measure  explained,  8 
Solids,  melting  points  of,  148 
Specific  heat  denned,  135 

162;  of  different  bodies,  1G3,  165, 

166 

—  heats  under  constant   pressure  and 
under  constant  volume,  164, 167 

—  gravities,  tables  of,  165 

of  oxygen  and  carbonic  acid, 

175 

Speed  of  steamers,  rule  for  determining, 
77 

steam  vessels,  how  to  determine, 

430,  432,  443 

vessels  a  main  condition  of  success 

in  war,  460 

common  steamers  may  be  increas- 
ed by  rocket  composition,  464 

Shafts,  strength  of,  133 

—  of  fly- wheel,  238.  239 

—  for  paddles,  294 ;  sizes  of  wronght-iron 
shafts  for  different  powers,  294 

Ships,  maximum  breadth  of,  best  posi- 
tion of,  417 

—  length  of,  should  vary  with  intended 
speed,  421 

—  resistance  of,  mainly  caused  by  fric- 
tion, 423,  442 

Spherical  measure,  9 

Spheroidal  condition  of  water,  109 

Square  measure  explained,  8 

—  and  circular  inches,  9 

—  roots,  nature  of,  44 
of  fractions,  45 

—  root,  method  of  extracting,  47 
Squares  and  square  roots,  44 

—  of  fractions,  45 

St.  Katharine's  Dock,  diagram  of  engine 

at,  346 

Standards  of  magnitude,  7 
Stanhope  press,  levers  of,  89 


SUN 

Stays  of  boilers,  321 

Steam-engines,  great  waste  of  heat  in. 
91 

Steam-engine,  theory  of  the,  134 

Steams,  latent  heats  of,  from  water,  al- 
cohol, ether,  and  sulphur  of  carbon, 
154 

Steam  and  water,  relative  bulks  of,  at  at- 
mospheric pressure,  102 

—  of  atmospheric  pressure,  density  of, 
102 

—  rushing  into  a  vacuum,  velocity  of, 
102 ;  velocity  the  same  at  all  pressures, 
102;    velocity  into    the  atmosphere, 
103 

—  sensible  and  latent  heat  of,  by  M. 
Regnault,  155;  elastic  force  of,  155- 
161 

—  expanding,  mean  pressure  of,  285 

—  ports,  216 

—  pipes,  proper  size  of,  218 

—  boilers,  proportions  of,  304 

—  room,  315 

—  ports  of  locomotives,  330 

—  pipes  of  locomotives,  331 

—  navigation,  399 

—  vessels,  locomotive  efficiency  of,  313 
Steamers,     equation    for     determining 

speed  of,  77 

Steamer '  Fairy,'  body  plan  of,  454 ;  '  Bat- 
tler,' 455;  'Bremen,'  456;  'Persia,' 
457;  '  Warrior,' 458 

Stones,  strength  of,  125 

—  machine  for  sawing,  889 

Strains  of  machines,  how  measured,  81, 
86 

—  infinite,  how  produced,  89 

Strap  of  side  rod,  proper  dimensions  of, 
259 

connecting  rod,  proper  dimensions 

of,  265 

Straps  of  air-pump  side  rods,  285 

Strengths  of  machines,  how  determined, 
81,  S6 

Strength  of  main  beam  of  an  engine, 
87 

of  materials,  124;  elastic  strength, 

124 

cast-iron  columns,  129, 131 ;  of  cast- 
iron  beams,  133 ;  of  shafts,  133 

boiler  to  withstand  any  given  pres- 
sure, 322 

Studs  of  the  beams  of  land  engines, 
232 

—  in  side  lever,  269 ;  metal  round  studs, 
271 

Subtraction,  nature  of,  13 ;  indicated  by 
—  or  minus,  14;  method  of  perform- 
ing, 15;  examples  of,  16 

—  of  fractions,  84 
Sugar  mill,  890 

Sun  the  source  of  mechanical  power, 
79 


INDEX. 


473 


SUP 

Superficial  measure  explained,  7 
Superheater,  proportions  of,  in  steamer 

'Khone,'3S3 

Surds  or  incommensurables,  46 
Surface  of  boiler  required  to  evaporate  a 

cubic  foot  of  water  per  hour,  309 

—  condensers,  proportions  of,  in  steamer 
'  Hansa,'  814 

—  condensers,  315 

—  heating,  of  modern  boilers,  375 

—  condensers  cause  internal  corrosion  in 
boilers,  331 :  proportions  of,  in  steamer 
'  Khone,'  3S8 


rp ABLE  of  addition,  11 
1     Tables,  multiplication,  19,  28 
'Tay1  steamer,  dimensions  of,  2S7 
Temperature  defined,  135 
Temperatures  of  liquefaction  and  ebulli- 
tion constant,  137 

steam  at  different  pressures,  159 

Tensile  strengths    of   metals,  126 ;    of 

woods,   127;    crushing   strengths  of 

woods,  128 ;  iron,  129 
—  strength  of  boiler  plates,  321 
'  Teviof  steamer,  dimensions  of,  287 
Theory  of  the  steam-engine,  134 
Thermo-dynamics,  134 
Thermometers,  187;  Centigrade,  Eeau- 

mur's.   and    Fahrenheit's  compared, 

139 

Thermal  unit,  162 
Thomson,  D.,  rotative  pumping  engines 

by,  862;  double  cylinder  engines  by, 

862;   combined  plunger  and  bucket 

pump  by,  363 

Throttling  the  steam,  effect  of,  198 
Time  during  which  bodies  have  fallen 

de terminable  from  their  velocity,  99 
by  height  fallen 

through,  98 
Toothed  wheels,  proportions  proper  for, 

246  . 

Torsion,  strength  to  resist,  of  different 

metals,  133 
Transverse  section  of  ships,  best  form  of, 

409 

Tubes  of  locomotive  boilers,  830 
'Tweed1  steamer,  dimensions  of,  287 
Tylor,  pumps  by,  885 


'TTLSTER,'  Indicator  diagrams  from, 

U     842,  345,  852 
Unit,  meaning  of  the  term,  5 
—  of  heat,  162 
Uptake  of  boilers,  sectional  area  of,  876 


TACTJUM,  velocity  of  air,  water,  and 
mercury  into.  101:  of  steam  and 
gas,  102 


WAG 

Values  of  different  coals  in  generating 

steam,  177 

Yalve  piston,  by  D.  Thomson,  363 
Vaporisation,  152 ;  latent  heat  of,  154 
Vapours  and  gases,  difference  between 

Velocities,  virtual,  law  of,  79 

Velocity  of  falling  bodies,  95 

determinate  from  height 

fallen,  97 ;  from  time  of  falling,  97 

air,  water,  and  mercury  into  a  va- 
cuum. 101 ;  of  steam  and  gas,  102 

rotation  that  will  burst  by  centrifu- 
gal force,  110 

—  permissible  in  bearings  moving  un- 
der a  given  pressure,  123 

water  in  rivers,  canals,  and  pipes, 

199 
water  flowing  in  pipes  and  canals, 

433 

Veneer  saw,  389 
Vermicelli  machine,  888 
Vertical  tubes,  advantages  of,  377 
Vessels,  resistance  of,  399 ;  proper  shape 

of,  401 

—  maximum  breadth  of,  best  position 
of,  417 

—  length  of,  should  vary  with  intended 
speed,  421 

—  resistance  of,  mainly  caused  by  fric- 
tion. 423,  442 

Vibrations  of  pendulums,  rule  for  deter- 
mining, 115 

'  Victoria  and  Albert,'  indicator  diagram 
from,  352 

Virtual  velocities,  law  of,  79 

Viscosity  or  molecular  attraction,  163 

Vis  ulna,  nature  of,  90 

Volumes,  relative,  of  water  and  steam  at 
atmospheric  pressure,  102 

—  and  pressures  of  gases,  146 

—  of  gases,  166 

Vulgar  fractions,  nature  of,  5 


"ITTAGON  boilers,  proportions  of.  810 
VV     "Water,  relative  density  of,  100 

—  into  a  vacuum,  velocity  of,  101 

—  and  steam,  relative  bulks  of,  at  atmos 
phcric  pressure.  102 

—  maximum  density  of,  189 

—  weight  of,  at  82%  189 

—  velocity   of,  In   rivers,  canals,    and 
pipes,  433 

—  works,  indicator  diagram  from  pump, 
861 

—  lines  of  ships,  400  ;    illustrated  by 
phape  of  fishes,  408 

—  in  pipes,  friction  of  the  same  at  all 
pressures,  42d 

—  velocity  of,  in  pipes,  434 ;  In  canals, 
434 


474 


INDEX. 


WAE 

"War,  maritime,  new  resources  available 
for,  401-464 

'Warrior'  steamer,  body  plan  of,  453; 
transverse  section  of,  459 

"Waste  water  pipe,  to  find  the  proper  di- 
ameter of,  224 

Wave  raised  by  a  vessel,  414, 419 ;  mo- 
tion of,  420 

Weaving  by  steam,  393;  by  compressed 
air,  398 

Weights  lifted  by  machines,  82 

Wenhanrs  double  cylinder  engine,  308 


ZIIR 

Wheels,  teeth  of,  24T 
Winch  wcijrhts  lifted  by,  SI 
Wirtx.'s  Zurich  machine,  3S6 
"Woods,  strength  of,  125 
Wool-spinning  mill,  8!)3 
Working  beam  of  land  engines,  how  to 
proportion,  233 


ZEKO,  absolute,  1ST 
Zurich  machine,  3S6 


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IJCSB    LIBRA*** 


A     000618778     5 


